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Properties

Label 14-1224e7-1.1-c3e7-0-1
Degree 1414
Conductor 4.116×10214.116\times 10^{21}
Sign 11
Analytic cond. 1.02454×10131.02454\times 10^{13}
Root an. cond. 8.498138.49813
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 26·7-s + 65·11-s + 17·13-s + 119·17-s + 125·19-s − 107·23-s − 345·25-s + 6·29-s + 44·31-s + 78·35-s + 254·37-s − 219·41-s + 419·43-s − 300·47-s − 277·49-s − 284·53-s + 195·55-s + 46·59-s + 638·61-s + 51·65-s + 796·67-s + 1.47e3·71-s + 1.71e3·73-s + 1.69e3·77-s + 1.13e3·79-s + 596·83-s + ⋯
L(s)  = 1  + 0.268·5-s + 1.40·7-s + 1.78·11-s + 0.362·13-s + 1.69·17-s + 1.50·19-s − 0.970·23-s − 2.75·25-s + 0.0384·29-s + 0.254·31-s + 0.376·35-s + 1.12·37-s − 0.834·41-s + 1.48·43-s − 0.931·47-s − 0.807·49-s − 0.736·53-s + 0.478·55-s + 0.101·59-s + 1.33·61-s + 0.0973·65-s + 1.45·67-s + 2.46·71-s + 2.74·73-s + 2.50·77-s + 1.61·79-s + 0.788·83-s + ⋯

Functional equation

Λ(s)=((221⋅314⋅177)s/2 ΓC(s)7 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 3^{14} \cdot 17^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((221⋅314⋅177)s/2 ΓC(s+3/2)7 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 3^{14} \cdot 17^{7}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1414
Conductor: 221⋅314⋅1772^{21} \cdot 3^{14} \cdot 17^{7}
Sign: 11
Analytic conductor: 1.02454×10131.02454\times 10^{13}
Root analytic conductor: 8.498138.49813
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (14, 221⋅314⋅177, ( :[3/2]7), 1)(14,\ 2^{21} \cdot 3^{14} \cdot 17^{7} ,\ ( \ : [3/2]^{7} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 53.8721178153.87211781
L(12)L(\frac12) ≈\approx 53.8721178153.87211781
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
17 (1−pT)7 ( 1 - p T )^{7}
good5 1−3T+354T2−107pT3+81988T4−120457T5+2699037pT6−88678p3T7+2699037p4T8−120457p6T9+81988p9T10−107p13T11+354p15T12−3p18T13+p21T14 1 - 3 T + 354 T^{2} - 107 p T^{3} + 81988 T^{4} - 120457 T^{5} + 2699037 p T^{6} - 88678 p^{3} T^{7} + 2699037 p^{4} T^{8} - 120457 p^{6} T^{9} + 81988 p^{9} T^{10} - 107 p^{13} T^{11} + 354 p^{15} T^{12} - 3 p^{18} T^{13} + p^{21} T^{14}
7 1−26T+953T2−15508T3+36471pT4−947054T5−38567559T6+1035613336T7−38567559p3T8−947054p6T9+36471p10T10−15508p12T11+953p15T12−26p18T13+p21T14 1 - 26 T + 953 T^{2} - 15508 T^{3} + 36471 p T^{4} - 947054 T^{5} - 38567559 T^{6} + 1035613336 T^{7} - 38567559 p^{3} T^{8} - 947054 p^{6} T^{9} + 36471 p^{10} T^{10} - 15508 p^{12} T^{11} + 953 p^{15} T^{12} - 26 p^{18} T^{13} + p^{21} T^{14}
11 1−65T+5460T2−199745T3+11724094T4−396579979T5+22768443575T6−700532534822T7+22768443575p3T8−396579979p6T9+11724094p9T10−199745p12T11+5460p15T12−65p18T13+p21T14 1 - 65 T + 5460 T^{2} - 199745 T^{3} + 11724094 T^{4} - 396579979 T^{5} + 22768443575 T^{6} - 700532534822 T^{7} + 22768443575 p^{3} T^{8} - 396579979 p^{6} T^{9} + 11724094 p^{9} T^{10} - 199745 p^{12} T^{11} + 5460 p^{15} T^{12} - 65 p^{18} T^{13} + p^{21} T^{14}
13 1−17T+3566T2−155977T3+13169472T4−445736523T5+31584320725T6−1462127060726T7+31584320725p3T8−445736523p6T9+13169472p9T10−155977p12T11+3566p15T12−17p18T13+p21T14 1 - 17 T + 3566 T^{2} - 155977 T^{3} + 13169472 T^{4} - 445736523 T^{5} + 31584320725 T^{6} - 1462127060726 T^{7} + 31584320725 p^{3} T^{8} - 445736523 p^{6} T^{9} + 13169472 p^{9} T^{10} - 155977 p^{12} T^{11} + 3566 p^{15} T^{12} - 17 p^{18} T^{13} + p^{21} T^{14}
19 1−125T+32564T2−3925317T3+536291318T4−57835084551T5+5510555058279T6−503257770414814T7+5510555058279p3T8−57835084551p6T9+536291318p9T10−3925317p12T11+32564p15T12−125p18T13+p21T14 1 - 125 T + 32564 T^{2} - 3925317 T^{3} + 536291318 T^{4} - 57835084551 T^{5} + 5510555058279 T^{6} - 503257770414814 T^{7} + 5510555058279 p^{3} T^{8} - 57835084551 p^{6} T^{9} + 536291318 p^{9} T^{10} - 3925317 p^{12} T^{11} + 32564 p^{15} T^{12} - 125 p^{18} T^{13} + p^{21} T^{14}
23 1+107T+53940T2+4243039T3+1375470422T4+87462702521T5+23069079907751T6+1240155430385582T7+23069079907751p3T8+87462702521p6T9+1375470422p9T10+4243039p12T11+53940p15T12+107p18T13+p21T14 1 + 107 T + 53940 T^{2} + 4243039 T^{3} + 1375470422 T^{4} + 87462702521 T^{5} + 23069079907751 T^{6} + 1240155430385582 T^{7} + 23069079907751 p^{3} T^{8} + 87462702521 p^{6} T^{9} + 1375470422 p^{9} T^{10} + 4243039 p^{12} T^{11} + 53940 p^{15} T^{12} + 107 p^{18} T^{13} + p^{21} T^{14}
29 1−6T+76371T2−5151988T3+2925956401T4−304123775298T5+90874281998963T6−8870279562105832T7+90874281998963p3T8−304123775298p6T9+2925956401p9T10−5151988p12T11+76371p15T12−6p18T13+p21T14 1 - 6 T + 76371 T^{2} - 5151988 T^{3} + 2925956401 T^{4} - 304123775298 T^{5} + 90874281998963 T^{6} - 8870279562105832 T^{7} + 90874281998963 p^{3} T^{8} - 304123775298 p^{6} T^{9} + 2925956401 p^{9} T^{10} - 5151988 p^{12} T^{11} + 76371 p^{15} T^{12} - 6 p^{18} T^{13} + p^{21} T^{14}
31 1−44T+49205T2−1421992T3+1232792589T4−133784265156T5+36888446996425T6−8534897219439056T7+36888446996425p3T8−133784265156p6T9+1232792589p9T10−1421992p12T11+49205p15T12−44p18T13+p21T14 1 - 44 T + 49205 T^{2} - 1421992 T^{3} + 1232792589 T^{4} - 133784265156 T^{5} + 36888446996425 T^{6} - 8534897219439056 T^{7} + 36888446996425 p^{3} T^{8} - 133784265156 p^{6} T^{9} + 1232792589 p^{9} T^{10} - 1421992 p^{12} T^{11} + 49205 p^{15} T^{12} - 44 p^{18} T^{13} + p^{21} T^{14}
37 1−254T+152571T2−26631556T3+12380828641T4−1512128030218T5+667910390260651T6−66410541275692936T7+667910390260651p3T8−1512128030218p6T9+12380828641p9T10−26631556p12T11+152571p15T12−254p18T13+p21T14 1 - 254 T + 152571 T^{2} - 26631556 T^{3} + 12380828641 T^{4} - 1512128030218 T^{5} + 667910390260651 T^{6} - 66410541275692936 T^{7} + 667910390260651 p^{3} T^{8} - 1512128030218 p^{6} T^{9} + 12380828641 p^{9} T^{10} - 26631556 p^{12} T^{11} + 152571 p^{15} T^{12} - 254 p^{18} T^{13} + p^{21} T^{14}
41 1+219T+185186T2+55111939T3+21969995724T4+6592395348305T5+2189011091591209T6+516453459461183394T7+2189011091591209p3T8+6592395348305p6T9+21969995724p9T10+55111939p12T11+185186p15T12+219p18T13+p21T14 1 + 219 T + 185186 T^{2} + 55111939 T^{3} + 21969995724 T^{4} + 6592395348305 T^{5} + 2189011091591209 T^{6} + 516453459461183394 T^{7} + 2189011091591209 p^{3} T^{8} + 6592395348305 p^{6} T^{9} + 21969995724 p^{9} T^{10} + 55111939 p^{12} T^{11} + 185186 p^{15} T^{12} + 219 p^{18} T^{13} + p^{21} T^{14}
43 1−419T+538868T2−171088339T3+121736298046T4−30470845203665T5+15529153543662711T6−3103403003686933362T7+15529153543662711p3T8−30470845203665p6T9+121736298046p9T10−171088339p12T11+538868p15T12−419p18T13+p21T14 1 - 419 T + 538868 T^{2} - 171088339 T^{3} + 121736298046 T^{4} - 30470845203665 T^{5} + 15529153543662711 T^{6} - 3103403003686933362 T^{7} + 15529153543662711 p^{3} T^{8} - 30470845203665 p^{6} T^{9} + 121736298046 p^{9} T^{10} - 171088339 p^{12} T^{11} + 538868 p^{15} T^{12} - 419 p^{18} T^{13} + p^{21} T^{14}
47 1+300T+340969T2+35379512T3+44599263157T4+1380783473428T5+5820010719291693T6+319026518445721104T7+5820010719291693p3T8+1380783473428p6T9+44599263157p9T10+35379512p12T11+340969p15T12+300p18T13+p21T14 1 + 300 T + 340969 T^{2} + 35379512 T^{3} + 44599263157 T^{4} + 1380783473428 T^{5} + 5820010719291693 T^{6} + 319026518445721104 T^{7} + 5820010719291693 p^{3} T^{8} + 1380783473428 p^{6} T^{9} + 44599263157 p^{9} T^{10} + 35379512 p^{12} T^{11} + 340969 p^{15} T^{12} + 300 p^{18} T^{13} + p^{21} T^{14}
53 1+284T+338771T2−16036792T3+45743266021T4−10691740383356T5+11466639834495487T6−466766187276585232T7+11466639834495487p3T8−10691740383356p6T9+45743266021p9T10−16036792p12T11+338771p15T12+284p18T13+p21T14 1 + 284 T + 338771 T^{2} - 16036792 T^{3} + 45743266021 T^{4} - 10691740383356 T^{5} + 11466639834495487 T^{6} - 466766187276585232 T^{7} + 11466639834495487 p^{3} T^{8} - 10691740383356 p^{6} T^{9} + 45743266021 p^{9} T^{10} - 16036792 p^{12} T^{11} + 338771 p^{15} T^{12} + 284 p^{18} T^{13} + p^{21} T^{14}
59 1−46T+661433T2−14945332T3+220177235209T4+2689413526270T5+55620054516264137T6+1563781907253732712T7+55620054516264137p3T8+2689413526270p6T9+220177235209p9T10−14945332p12T11+661433p15T12−46p18T13+p21T14 1 - 46 T + 661433 T^{2} - 14945332 T^{3} + 220177235209 T^{4} + 2689413526270 T^{5} + 55620054516264137 T^{6} + 1563781907253732712 T^{7} + 55620054516264137 p^{3} T^{8} + 2689413526270 p^{6} T^{9} + 220177235209 p^{9} T^{10} - 14945332 p^{12} T^{11} + 661433 p^{15} T^{12} - 46 p^{18} T^{13} + p^{21} T^{14}
61 1−638T+796371T2−259635300T3+284706012689T4−83685460081450T5+91585740973986003T6−24629039672906966472T7+91585740973986003p3T8−83685460081450p6T9+284706012689p9T10−259635300p12T11+796371p15T12−638p18T13+p21T14 1 - 638 T + 796371 T^{2} - 259635300 T^{3} + 284706012689 T^{4} - 83685460081450 T^{5} + 91585740973986003 T^{6} - 24629039672906966472 T^{7} + 91585740973986003 p^{3} T^{8} - 83685460081450 p^{6} T^{9} + 284706012689 p^{9} T^{10} - 259635300 p^{12} T^{11} + 796371 p^{15} T^{12} - 638 p^{18} T^{13} + p^{21} T^{14}
67 1−796T+843445T2−474463576T3+434669122405T4−239244171464260T5+180700031996011913T6−84885796738878461648T7+180700031996011913p3T8−239244171464260p6T9+434669122405p9T10−474463576p12T11+843445p15T12−796p18T13+p21T14 1 - 796 T + 843445 T^{2} - 474463576 T^{3} + 434669122405 T^{4} - 239244171464260 T^{5} + 180700031996011913 T^{6} - 84885796738878461648 T^{7} + 180700031996011913 p^{3} T^{8} - 239244171464260 p^{6} T^{9} + 434669122405 p^{9} T^{10} - 474463576 p^{12} T^{11} + 843445 p^{15} T^{12} - 796 p^{18} T^{13} + p^{21} T^{14}
71 1−1472T+1895245T2−1489924640T3+1058612301997T4−531767222771328T5+281647459552971457T6− 1 - 1472 T + 1895245 T^{2} - 1489924640 T^{3} + 1058612301997 T^{4} - 531767222771328 T^{5} + 281647459552971457 T^{6} - 13 ⁣⋯ ⁣4013\!\cdots\!40T7+281647459552971457p3T8−531767222771328p6T9+1058612301997p9T10−1489924640p12T11+1895245p15T12−1472p18T13+p21T14 T^{7} + 281647459552971457 p^{3} T^{8} - 531767222771328 p^{6} T^{9} + 1058612301997 p^{9} T^{10} - 1489924640 p^{12} T^{11} + 1895245 p^{15} T^{12} - 1472 p^{18} T^{13} + p^{21} T^{14}
73 1−1712T+2844231T2−2853363072T3+2641240205581T4−1911511071335504T5+1324276635663873627T6− 1 - 1712 T + 2844231 T^{2} - 2853363072 T^{3} + 2641240205581 T^{4} - 1911511071335504 T^{5} + 1324276635663873627 T^{6} - 82 ⁣⋯ ⁣3282\!\cdots\!32T7+1324276635663873627p3T8−1911511071335504p6T9+2641240205581p9T10−2853363072p12T11+2844231p15T12−1712p18T13+p21T14 T^{7} + 1324276635663873627 p^{3} T^{8} - 1911511071335504 p^{6} T^{9} + 2641240205581 p^{9} T^{10} - 2853363072 p^{12} T^{11} + 2844231 p^{15} T^{12} - 1712 p^{18} T^{13} + p^{21} T^{14}
79 1−1134T+1652497T2−995418940T3+1245056154849T4−761460794061514T5+864315482858125905T6− 1 - 1134 T + 1652497 T^{2} - 995418940 T^{3} + 1245056154849 T^{4} - 761460794061514 T^{5} + 864315482858125905 T^{6} - 45 ⁣⋯ ⁣8445\!\cdots\!84T7+864315482858125905p3T8−761460794061514p6T9+1245056154849p9T10−995418940p12T11+1652497p15T12−1134p18T13+p21T14 T^{7} + 864315482858125905 p^{3} T^{8} - 761460794061514 p^{6} T^{9} + 1245056154849 p^{9} T^{10} - 995418940 p^{12} T^{11} + 1652497 p^{15} T^{12} - 1134 p^{18} T^{13} + p^{21} T^{14}
83 1−596T+1446597T2+59437368T3+846468515397T4+243263035026356T5+803333180124238777T6+12916505499420261648T7+803333180124238777p3T8+243263035026356p6T9+846468515397p9T10+59437368p12T11+1446597p15T12−596p18T13+p21T14 1 - 596 T + 1446597 T^{2} + 59437368 T^{3} + 846468515397 T^{4} + 243263035026356 T^{5} + 803333180124238777 T^{6} + 12916505499420261648 T^{7} + 803333180124238777 p^{3} T^{8} + 243263035026356 p^{6} T^{9} + 846468515397 p^{9} T^{10} + 59437368 p^{12} T^{11} + 1446597 p^{15} T^{12} - 596 p^{18} T^{13} + p^{21} T^{14}
89 1+694T+2622227T2+1063809756T3+3483368838233T4+1079225771839434T5+3386971173433246083T6+ 1 + 694 T + 2622227 T^{2} + 1063809756 T^{3} + 3483368838233 T^{4} + 1079225771839434 T^{5} + 3386971173433246083 T^{6} + 91 ⁣⋯ ⁣0091\!\cdots\!00T7+3386971173433246083p3T8+1079225771839434p6T9+3483368838233p9T10+1063809756p12T11+2622227p15T12+694p18T13+p21T14 T^{7} + 3386971173433246083 p^{3} T^{8} + 1079225771839434 p^{6} T^{9} + 3483368838233 p^{9} T^{10} + 1063809756 p^{12} T^{11} + 2622227 p^{15} T^{12} + 694 p^{18} T^{13} + p^{21} T^{14}
97 1−2536T+7635023T2−12963867984T3+22468525613293T4−28310220541550360T5+35083068932946523139T6− 1 - 2536 T + 7635023 T^{2} - 12963867984 T^{3} + 22468525613293 T^{4} - 28310220541550360 T^{5} + 35083068932946523139 T^{6} - 33 ⁣⋯ ⁣1233\!\cdots\!12T7+35083068932946523139p3T8−28310220541550360p6T9+22468525613293p9T10−12963867984p12T11+7635023p15T12−2536p18T13+p21T14 T^{7} + 35083068932946523139 p^{3} T^{8} - 28310220541550360 p^{6} T^{9} + 22468525613293 p^{9} T^{10} - 12963867984 p^{12} T^{11} + 7635023 p^{15} T^{12} - 2536 p^{18} T^{13} + p^{21} T^{14}
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   L(s)=∏p ∏j=114(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.21595960043040141022716850372, −4.06740319057999964056686889548, −3.87245850712892067336706079619, −3.83519101110307237682952681424, −3.50077139789920848953201241817, −3.38123459280745821858195512929, −3.34877416734750132792952413243, −3.18934545597494939537925709794, −3.00752112668854588758544391002, −2.90533768314707613129636335503, −2.90508832094880162166990744212, −2.16053694530025144304246942149, −2.15284170326904767680885823863, −2.05365447469452497049999579601, −2.02694975754609744388706509557, −1.87300580938177089986096016440, −1.81629275648447273797289424441, −1.58232496752584933811990733923, −1.35341730592656442427451010955, −0.894630577938067830945599931172, −0.862499973278605173857352109564, −0.74555441366124507945067814487, −0.71337030059210514093489454299, −0.59037598130494062200921180040, −0.26262498320072904688911710252, 0.26262498320072904688911710252, 0.59037598130494062200921180040, 0.71337030059210514093489454299, 0.74555441366124507945067814487, 0.862499973278605173857352109564, 0.894630577938067830945599931172, 1.35341730592656442427451010955, 1.58232496752584933811990733923, 1.81629275648447273797289424441, 1.87300580938177089986096016440, 2.02694975754609744388706509557, 2.05365447469452497049999579601, 2.15284170326904767680885823863, 2.16053694530025144304246942149, 2.90508832094880162166990744212, 2.90533768314707613129636335503, 3.00752112668854588758544391002, 3.18934545597494939537925709794, 3.34877416734750132792952413243, 3.38123459280745821858195512929, 3.50077139789920848953201241817, 3.83519101110307237682952681424, 3.87245850712892067336706079619, 4.06740319057999964056686889548, 4.21595960043040141022716850372

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.