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Properties

Label 12-425e6-1.1-c3e6-0-4
Degree 1212
Conductor 5.893×10155.893\times 10^{15}
Sign 11
Analytic cond. 2.48616×1082.48616\times 10^{8}
Root an. cond. 5.007575.00757
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 66

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 6·3-s − 17·4-s − 6·6-s − 4·7-s + 25·8-s − 69·9-s − 20·11-s − 102·12-s − 42·13-s + 4·14-s + 78·16-s − 102·17-s + 69·18-s − 70·19-s − 24·21-s + 20·22-s + 92·23-s + 150·24-s + 42·26-s − 414·27-s + 68·28-s − 316·29-s − 758·31-s − 174·32-s − 120·33-s + 102·34-s + ⋯
L(s)  = 1  − 0.353·2-s + 1.15·3-s − 2.12·4-s − 0.408·6-s − 0.215·7-s + 1.10·8-s − 2.55·9-s − 0.548·11-s − 2.45·12-s − 0.896·13-s + 0.0763·14-s + 1.21·16-s − 1.45·17-s + 0.903·18-s − 0.845·19-s − 0.249·21-s + 0.193·22-s + 0.834·23-s + 1.27·24-s + 0.316·26-s − 2.95·27-s + 0.458·28-s − 2.02·29-s − 4.39·31-s − 0.961·32-s − 0.633·33-s + 0.514·34-s + ⋯

Functional equation

Λ(s)=((512⋅176)s/2 ΓC(s)6 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((512⋅176)s/2 ΓC(s+3/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 512⋅1765^{12} \cdot 17^{6}
Sign: 11
Analytic conductor: 2.48616×1082.48616\times 10^{8}
Root analytic conductor: 5.007575.00757
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 66
Selberg data: (12, 512⋅176, ( :[3/2]6), 1)(12,\ 5^{12} \cdot 17^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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)
bad5 1 1
17 (1+pT)6 ( 1 + p T )^{6}
good2 1+T+9pT2+5pT3+213T4+29T5+463p2T6+29p3T7+213p6T8+5p10T9+9p13T10+p15T11+p18T12 1 + T + 9 p T^{2} + 5 p T^{3} + 213 T^{4} + 29 T^{5} + 463 p^{2} T^{6} + 29 p^{3} T^{7} + 213 p^{6} T^{8} + 5 p^{10} T^{9} + 9 p^{13} T^{10} + p^{15} T^{11} + p^{18} T^{12}
3 1−2pT+35pT2−70p2T3+1904pT4−10144pT5+189869T6−10144p4T7+1904p7T8−70p11T9+35p13T10−2p16T11+p18T12 1 - 2 p T + 35 p T^{2} - 70 p^{2} T^{3} + 1904 p T^{4} - 10144 p T^{5} + 189869 T^{6} - 10144 p^{4} T^{7} + 1904 p^{7} T^{8} - 70 p^{11} T^{9} + 35 p^{13} T^{10} - 2 p^{16} T^{11} + p^{18} T^{12}
7 1+4T+1079T2+1168T3+94062pT4+98386T5+266269953T6+98386p3T7+94062p7T8+1168p9T9+1079p12T10+4p15T11+p18T12 1 + 4 T + 1079 T^{2} + 1168 T^{3} + 94062 p T^{4} + 98386 T^{5} + 266269953 T^{6} + 98386 p^{3} T^{7} + 94062 p^{7} T^{8} + 1168 p^{9} T^{9} + 1079 p^{12} T^{10} + 4 p^{15} T^{11} + p^{18} T^{12}
11 1+20T+4948T2+43012T3+10814227T4+11855292T5+15873719200T6+11855292p3T7+10814227p6T8+43012p9T9+4948p12T10+20p15T11+p18T12 1 + 20 T + 4948 T^{2} + 43012 T^{3} + 10814227 T^{4} + 11855292 T^{5} + 15873719200 T^{6} + 11855292 p^{3} T^{7} + 10814227 p^{6} T^{8} + 43012 p^{9} T^{9} + 4948 p^{12} T^{10} + 20 p^{15} T^{11} + p^{18} T^{12}
13 1+42T+7869T2+151042T3+24727174T4+111870518T5+54714903749T6+111870518p3T7+24727174p6T8+151042p9T9+7869p12T10+42p15T11+p18T12 1 + 42 T + 7869 T^{2} + 151042 T^{3} + 24727174 T^{4} + 111870518 T^{5} + 54714903749 T^{6} + 111870518 p^{3} T^{7} + 24727174 p^{6} T^{8} + 151042 p^{9} T^{9} + 7869 p^{12} T^{10} + 42 p^{15} T^{11} + p^{18} T^{12}
19 1+70T+26826T2+2464698T3+330431959T4+34215767068T5+2653157592748T6+34215767068p3T7+330431959p6T8+2464698p9T9+26826p12T10+70p15T11+p18T12 1 + 70 T + 26826 T^{2} + 2464698 T^{3} + 330431959 T^{4} + 34215767068 T^{5} + 2653157592748 T^{6} + 34215767068 p^{3} T^{7} + 330431959 p^{6} T^{8} + 2464698 p^{9} T^{9} + 26826 p^{12} T^{10} + 70 p^{15} T^{11} + p^{18} T^{12}
23 1−4pT+42788T2−5436848T3+830575523T4−129470295824T5+11130729824624T6−129470295824p3T7+830575523p6T8−5436848p9T9+42788p12T10−4p16T11+p18T12 1 - 4 p T + 42788 T^{2} - 5436848 T^{3} + 830575523 T^{4} - 129470295824 T^{5} + 11130729824624 T^{6} - 129470295824 p^{3} T^{7} + 830575523 p^{6} T^{8} - 5436848 p^{9} T^{9} + 42788 p^{12} T^{10} - 4 p^{16} T^{11} + p^{18} T^{12}
29 1+316T+109526T2+23100548T3+4747718199T4+823410852832T5+133168996602228T6+823410852832p3T7+4747718199p6T8+23100548p9T9+109526p12T10+316p15T11+p18T12 1 + 316 T + 109526 T^{2} + 23100548 T^{3} + 4747718199 T^{4} + 823410852832 T^{5} + 133168996602228 T^{6} + 823410852832 p^{3} T^{7} + 4747718199 p^{6} T^{8} + 23100548 p^{9} T^{9} + 109526 p^{12} T^{10} + 316 p^{15} T^{11} + p^{18} T^{12}
31 1+758T+360977T2+123118754T3+33543209988T4+7472355028736T5+1406393966922489T6+7472355028736p3T7+33543209988p6T8+123118754p9T9+360977p12T10+758p15T11+p18T12 1 + 758 T + 360977 T^{2} + 123118754 T^{3} + 33543209988 T^{4} + 7472355028736 T^{5} + 1406393966922489 T^{6} + 7472355028736 p^{3} T^{7} + 33543209988 p^{6} T^{8} + 123118754 p^{9} T^{9} + 360977 p^{12} T^{10} + 758 p^{15} T^{11} + p^{18} T^{12}
37 1+76T+169906T2+10502668T3+15896846071T4+745784974728T5+956093792902748T6+745784974728p3T7+15896846071p6T8+10502668p9T9+169906p12T10+76p15T11+p18T12 1 + 76 T + 169906 T^{2} + 10502668 T^{3} + 15896846071 T^{4} + 745784974728 T^{5} + 956093792902748 T^{6} + 745784974728 p^{3} T^{7} + 15896846071 p^{6} T^{8} + 10502668 p^{9} T^{9} + 169906 p^{12} T^{10} + 76 p^{15} T^{11} + p^{18} T^{12}
41 1+512T+364466T2+138040792T3+58919063775T4+17026052439928T5+5298258739038460T6+17026052439928p3T7+58919063775p6T8+138040792p9T9+364466p12T10+512p15T11+p18T12 1 + 512 T + 364466 T^{2} + 138040792 T^{3} + 58919063775 T^{4} + 17026052439928 T^{5} + 5298258739038460 T^{6} + 17026052439928 p^{3} T^{7} + 58919063775 p^{6} T^{8} + 138040792 p^{9} T^{9} + 364466 p^{12} T^{10} + 512 p^{15} T^{11} + p^{18} T^{12}
43 1+70T+197874T2−29168326T3+10826853927T4−7592990627284T5+302366159104508T6−7592990627284p3T7+10826853927p6T8−29168326p9T9+197874p12T10+70p15T11+p18T12 1 + 70 T + 197874 T^{2} - 29168326 T^{3} + 10826853927 T^{4} - 7592990627284 T^{5} + 302366159104508 T^{6} - 7592990627284 p^{3} T^{7} + 10826853927 p^{6} T^{8} - 29168326 p^{9} T^{9} + 197874 p^{12} T^{10} + 70 p^{15} T^{11} + p^{18} T^{12}
47 1−448T+580798T2−198993872T3+141567396175T4−37888800109264T5+19115530685467620T6−37888800109264p3T7+141567396175p6T8−198993872p9T9+580798p12T10−448p15T11+p18T12 1 - 448 T + 580798 T^{2} - 198993872 T^{3} + 141567396175 T^{4} - 37888800109264 T^{5} + 19115530685467620 T^{6} - 37888800109264 p^{3} T^{7} + 141567396175 p^{6} T^{8} - 198993872 p^{9} T^{9} + 580798 p^{12} T^{10} - 448 p^{15} T^{11} + p^{18} T^{12}
53 1+734T+847989T2+431012234T3+287021168218T4+110583765819462T5+54487722665556021T6+110583765819462p3T7+287021168218p6T8+431012234p9T9+847989p12T10+734p15T11+p18T12 1 + 734 T + 847989 T^{2} + 431012234 T^{3} + 287021168218 T^{4} + 110583765819462 T^{5} + 54487722665556021 T^{6} + 110583765819462 p^{3} T^{7} + 287021168218 p^{6} T^{8} + 431012234 p^{9} T^{9} + 847989 p^{12} T^{10} + 734 p^{15} T^{11} + p^{18} T^{12}
59 1+238T+995334T2+197942850T3+452027119815T4+73669147883468T5+118899853181909524T6+73669147883468p3T7+452027119815p6T8+197942850p9T9+995334p12T10+238p15T11+p18T12 1 + 238 T + 995334 T^{2} + 197942850 T^{3} + 452027119815 T^{4} + 73669147883468 T^{5} + 118899853181909524 T^{6} + 73669147883468 p^{3} T^{7} + 452027119815 p^{6} T^{8} + 197942850 p^{9} T^{9} + 995334 p^{12} T^{10} + 238 p^{15} T^{11} + p^{18} T^{12}
61 1+1188T+1748090T2+1367574628T3+1116299329335T4+624612049275464T5+350026644826364172T6+624612049275464p3T7+1116299329335p6T8+1367574628p9T9+1748090p12T10+1188p15T11+p18T12 1 + 1188 T + 1748090 T^{2} + 1367574628 T^{3} + 1116299329335 T^{4} + 624612049275464 T^{5} + 350026644826364172 T^{6} + 624612049275464 p^{3} T^{7} + 1116299329335 p^{6} T^{8} + 1367574628 p^{9} T^{9} + 1748090 p^{12} T^{10} + 1188 p^{15} T^{11} + p^{18} T^{12}
67 1−768T+706690T2−192601280T3+237734438775T4−127111011778624T5+127125428249077628T6−127111011778624p3T7+237734438775p6T8−192601280p9T9+706690p12T10−768p15T11+p18T12 1 - 768 T + 706690 T^{2} - 192601280 T^{3} + 237734438775 T^{4} - 127111011778624 T^{5} + 127125428249077628 T^{6} - 127111011778624 p^{3} T^{7} + 237734438775 p^{6} T^{8} - 192601280 p^{9} T^{9} + 706690 p^{12} T^{10} - 768 p^{15} T^{11} + p^{18} T^{12}
71 1+1276T+1911811T2+1562133684T3+1375832897030T4+868606251808150T5+592866246651136721T6+868606251808150p3T7+1375832897030p6T8+1562133684p9T9+1911811p12T10+1276p15T11+p18T12 1 + 1276 T + 1911811 T^{2} + 1562133684 T^{3} + 1375832897030 T^{4} + 868606251808150 T^{5} + 592866246651136721 T^{6} + 868606251808150 p^{3} T^{7} + 1375832897030 p^{6} T^{8} + 1562133684 p^{9} T^{9} + 1911811 p^{12} T^{10} + 1276 p^{15} T^{11} + p^{18} T^{12}
73 1+84T+1019274T2+8385356T3+625009606863T4+32327144607888T5+300051831193873292T6+32327144607888p3T7+625009606863p6T8+8385356p9T9+1019274p12T10+84p15T11+p18T12 1 + 84 T + 1019274 T^{2} + 8385356 T^{3} + 625009606863 T^{4} + 32327144607888 T^{5} + 300051831193873292 T^{6} + 32327144607888 p^{3} T^{7} + 625009606863 p^{6} T^{8} + 8385356 p^{9} T^{9} + 1019274 p^{12} T^{10} + 84 p^{15} T^{11} + p^{18} T^{12}
79 1+3066T+5975027T2+8147234562T3+8883522900366T4+7915792843234616T5+6033024976452237145T6+7915792843234616p3T7+8883522900366p6T8+8147234562p9T9+5975027p12T10+3066p15T11+p18T12 1 + 3066 T + 5975027 T^{2} + 8147234562 T^{3} + 8883522900366 T^{4} + 7915792843234616 T^{5} + 6033024976452237145 T^{6} + 7915792843234616 p^{3} T^{7} + 8883522900366 p^{6} T^{8} + 8147234562 p^{9} T^{9} + 5975027 p^{12} T^{10} + 3066 p^{15} T^{11} + p^{18} T^{12}
83 1+92T+1323682T2+273166884T3+1286946563863T4+191484652785656T5+860607993399136316T6+191484652785656p3T7+1286946563863p6T8+273166884p9T9+1323682p12T10+92p15T11+p18T12 1 + 92 T + 1323682 T^{2} + 273166884 T^{3} + 1286946563863 T^{4} + 191484652785656 T^{5} + 860607993399136316 T^{6} + 191484652785656 p^{3} T^{7} + 1286946563863 p^{6} T^{8} + 273166884 p^{9} T^{9} + 1323682 p^{12} T^{10} + 92 p^{15} T^{11} + p^{18} T^{12}
89 1+1760T+2798174T2+4169476384T3+4526987039919T4+4570352781182384T5+4300812649730173092T6+4570352781182384p3T7+4526987039919p6T8+4169476384p9T9+2798174p12T10+1760p15T11+p18T12 1 + 1760 T + 2798174 T^{2} + 4169476384 T^{3} + 4526987039919 T^{4} + 4570352781182384 T^{5} + 4300812649730173092 T^{6} + 4570352781182384 p^{3} T^{7} + 4526987039919 p^{6} T^{8} + 4169476384 p^{9} T^{9} + 2798174 p^{12} T^{10} + 1760 p^{15} T^{11} + p^{18} T^{12}
97 1−12pT+2791838T2−3926683212T3+5091862013775T4−5432759795084648T5+6177688080875556580T6−5432759795084648p3T7+5091862013775p6T8−3926683212p9T9+2791838p12T10−12p16T11+p18T12 1 - 12 p T + 2791838 T^{2} - 3926683212 T^{3} + 5091862013775 T^{4} - 5432759795084648 T^{5} + 6177688080875556580 T^{6} - 5432759795084648 p^{3} T^{7} + 5091862013775 p^{6} T^{8} - 3926683212 p^{9} T^{9} + 2791838 p^{12} T^{10} - 12 p^{16} T^{11} + p^{18} T^{12}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.22613386591412461296648561359, −5.88109819171209603335669000030, −5.57335889864923687374869103025, −5.38713727173116145507439401055, −5.30207028857165183710073217305, −5.25944823858072719605304131250, −5.24294161077458733030047651285, −4.89896620513056292980913755419, −4.56601367235211404810101891550, −4.40873618677941242235883295377, −4.33591653580664017420712198753, −4.04939618099608166281072170109, −4.03643332916246655663454454925, −3.55078489590976208503364549774, −3.46339446809314426930640943390, −3.20570749486415411899448185612, −3.10865382454852942656713926570, −2.88708930413310177293794103579, −2.68398862508619512758105295027, −2.34913138908517530977372910233, −2.31717791152293715593995312592, −1.93149043706285290329073491493, −1.48907858398747149608729112713, −1.48593948131549558195888611335, −1.28840677838444840479038845230, 0, 0, 0, 0, 0, 0, 1.28840677838444840479038845230, 1.48593948131549558195888611335, 1.48907858398747149608729112713, 1.93149043706285290329073491493, 2.31717791152293715593995312592, 2.34913138908517530977372910233, 2.68398862508619512758105295027, 2.88708930413310177293794103579, 3.10865382454852942656713926570, 3.20570749486415411899448185612, 3.46339446809314426930640943390, 3.55078489590976208503364549774, 4.03643332916246655663454454925, 4.04939618099608166281072170109, 4.33591653580664017420712198753, 4.40873618677941242235883295377, 4.56601367235211404810101891550, 4.89896620513056292980913755419, 5.24294161077458733030047651285, 5.25944823858072719605304131250, 5.30207028857165183710073217305, 5.38713727173116145507439401055, 5.57335889864923687374869103025, 5.88109819171209603335669000030, 6.22613386591412461296648561359

Graph of the ZZ-function along the critical line

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