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Properties

Label 12-448e6-1.1-c3e6-0-3
Degree 1212
Conductor 8.085×10158.085\times 10^{15}
Sign 11
Analytic cond. 3.41086×1083.41086\times 10^{8}
Root an. cond. 5.141285.14128
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  − 7·3-s − 3·5-s − 4·7-s + 56·9-s − 3·11-s + 52·13-s + 21·15-s + 31·17-s − 89·19-s + 28·21-s + 201·23-s + 42·25-s − 91·27-s − 380·29-s + 339·31-s + 21·33-s + 12·35-s − 535·37-s − 364·39-s + 116·41-s − 536·43-s − 168·45-s + 205·47-s − 757·49-s − 217·51-s + 757·53-s + 9·55-s + ⋯
L(s)  = 1  − 1.34·3-s − 0.268·5-s − 0.215·7-s + 2.07·9-s − 0.0822·11-s + 1.10·13-s + 0.361·15-s + 0.442·17-s − 1.07·19-s + 0.290·21-s + 1.82·23-s + 0.335·25-s − 0.648·27-s − 2.43·29-s + 1.96·31-s + 0.110·33-s + 0.0579·35-s − 2.37·37-s − 1.49·39-s + 0.441·41-s − 1.90·43-s − 0.556·45-s + 0.636·47-s − 2.20·49-s − 0.595·51-s + 1.96·53-s + 0.0220·55-s + ⋯

Functional equation

Λ(s)=((236⋅76)s/2 ΓC(s)6 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((236⋅76)s/2 ΓC(s+3/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 236⋅762^{36} \cdot 7^{6}
Sign: 11
Analytic conductor: 3.41086×1083.41086\times 10^{8}
Root analytic conductor: 5.141285.14128
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 236⋅76, ( :[3/2]6), 1)(12,\ 2^{36} \cdot 7^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 2.9617042592.961704259
L(12)L(\frac12) ≈\approx 2.9617042592.961704259
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
7 1+4T+773T2+328pT3+773p3T4+4p6T5+p9T6 1 + 4 T + 773 T^{2} + 328 p T^{3} + 773 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6}
good3 1+7T−7T2−350T3−973T4+4711T5+56614T6+4711p3T7−973p6T8−350p9T9−7p12T10+7p15T11+p18T12 1 + 7 T - 7 T^{2} - 350 T^{3} - 973 T^{4} + 4711 T^{5} + 56614 T^{6} + 4711 p^{3} T^{7} - 973 p^{6} T^{8} - 350 p^{9} T^{9} - 7 p^{12} T^{10} + 7 p^{15} T^{11} + p^{18} T^{12}
5 1+3T−33T2+2752T3+453T4−39771T5+5269094T6−39771p3T7+453p6T8+2752p9T9−33p12T10+3p15T11+p18T12 1 + 3 T - 33 T^{2} + 2752 T^{3} + 453 T^{4} - 39771 T^{5} + 5269094 T^{6} - 39771 p^{3} T^{7} + 453 p^{6} T^{8} + 2752 p^{9} T^{9} - 33 p^{12} T^{10} + 3 p^{15} T^{11} + p^{18} T^{12}
11 1+3T−1671T2−88206T3+446451T4+70224843T5+3117041062T6+70224843p3T7+446451p6T8−88206p9T9−1671p12T10+3p15T11+p18T12 1 + 3 T - 1671 T^{2} - 88206 T^{3} + 446451 T^{4} + 70224843 T^{5} + 3117041062 T^{6} + 70224843 p^{3} T^{7} + 446451 p^{6} T^{8} - 88206 p^{9} T^{9} - 1671 p^{12} T^{10} + 3 p^{15} T^{11} + p^{18} T^{12}
13 (1−2pT+4347T2−140572T3+4347p3T4−2p7T5+p9T6)2 ( 1 - 2 p T + 4347 T^{2} - 140572 T^{3} + 4347 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} )^{2}
17 1−31T−7341T2+5924pT3+22585393T4+294225803T5−92586620842T6+294225803p3T7+22585393p6T8+5924p10T9−7341p12T10−31p15T11+p18T12 1 - 31 T - 7341 T^{2} + 5924 p T^{3} + 22585393 T^{4} + 294225803 T^{5} - 92586620842 T^{6} + 294225803 p^{3} T^{7} + 22585393 p^{6} T^{8} + 5924 p^{10} T^{9} - 7341 p^{12} T^{10} - 31 p^{15} T^{11} + p^{18} T^{12}
19 1+89T−13039T2−531306T3+188901763T4+3714875569T5−1304484035114T6+3714875569p3T7+188901763p6T8−531306p9T9−13039p12T10+89p15T11+p18T12 1 + 89 T - 13039 T^{2} - 531306 T^{3} + 188901763 T^{4} + 3714875569 T^{5} - 1304484035114 T^{6} + 3714875569 p^{3} T^{7} + 188901763 p^{6} T^{8} - 531306 p^{9} T^{9} - 13039 p^{12} T^{10} + 89 p^{15} T^{11} + p^{18} T^{12}
23 1−201T−7035T2+419006T3+672679263T4−33819862221T5−4987613588554T6−33819862221p3T7+672679263p6T8+419006p9T9−7035p12T10−201p15T11+p18T12 1 - 201 T - 7035 T^{2} + 419006 T^{3} + 672679263 T^{4} - 33819862221 T^{5} - 4987613588554 T^{6} - 33819862221 p^{3} T^{7} + 672679263 p^{6} T^{8} + 419006 p^{9} T^{9} - 7035 p^{12} T^{10} - 201 p^{15} T^{11} + p^{18} T^{12}
29 (1+190T+81643T2+9219316T3+81643p3T4+190p6T5+p9T6)2 ( 1 + 190 T + 81643 T^{2} + 9219316 T^{3} + 81643 p^{3} T^{4} + 190 p^{6} T^{5} + p^{9} T^{6} )^{2}
31 1−339T+14469T2+1234882T3+1116704127T4−44284930023T5−38410082180298T6−44284930023p3T7+1116704127p6T8+1234882p9T9+14469p12T10−339p15T11+p18T12 1 - 339 T + 14469 T^{2} + 1234882 T^{3} + 1116704127 T^{4} - 44284930023 T^{5} - 38410082180298 T^{6} - 44284930023 p^{3} T^{7} + 1116704127 p^{6} T^{8} + 1234882 p^{9} T^{9} + 14469 p^{12} T^{10} - 339 p^{15} T^{11} + p^{18} T^{12}
37 1+535T+77743T2+4911224T3+118429129pT4+1188076376281T5+166239153232886T6+1188076376281p3T7+118429129p7T8+4911224p9T9+77743p12T10+535p15T11+p18T12 1 + 535 T + 77743 T^{2} + 4911224 T^{3} + 118429129 p T^{4} + 1188076376281 T^{5} + 166239153232886 T^{6} + 1188076376281 p^{3} T^{7} + 118429129 p^{7} T^{8} + 4911224 p^{9} T^{9} + 77743 p^{12} T^{10} + 535 p^{15} T^{11} + p^{18} T^{12}
41 (1−58T+164327T2−4134444T3+164327p3T4−58p6T5+p9T6)2 ( 1 - 58 T + 164327 T^{2} - 4134444 T^{3} + 164327 p^{3} T^{4} - 58 p^{6} T^{5} + p^{9} T^{6} )^{2}
43 (1+268T+132841T2+18272264T3+132841p3T4+268p6T5+p9T6)2 ( 1 + 268 T + 132841 T^{2} + 18272264 T^{3} + 132841 p^{3} T^{4} + 268 p^{6} T^{5} + p^{9} T^{6} )^{2}
47 1−205T−183403T2+16447246T3+20990523375T4+323493805591T5−2509221820128074T6+323493805591p3T7+20990523375p6T8+16447246p9T9−183403p12T10−205p15T11+p18T12 1 - 205 T - 183403 T^{2} + 16447246 T^{3} + 20990523375 T^{4} + 323493805591 T^{5} - 2509221820128074 T^{6} + 323493805591 p^{3} T^{7} + 20990523375 p^{6} T^{8} + 16447246 p^{9} T^{9} - 183403 p^{12} T^{10} - 205 p^{15} T^{11} + p^{18} T^{12}
53 1−757T+229743T2+41591296T3−47359243259T4+8853768931901T5−926599751279866T6+8853768931901p3T7−47359243259p6T8+41591296p9T9+229743p12T10−757p15T11+p18T12 1 - 757 T + 229743 T^{2} + 41591296 T^{3} - 47359243259 T^{4} + 8853768931901 T^{5} - 926599751279866 T^{6} + 8853768931901 p^{3} T^{7} - 47359243259 p^{6} T^{8} + 41591296 p^{9} T^{9} + 229743 p^{12} T^{10} - 757 p^{15} T^{11} + p^{18} T^{12}
59 1+1799T+1576225T2+1115416106T3+732038218331T4+402383977564751T5+190135753613171014T6+402383977564751p3T7+732038218331p6T8+1115416106p9T9+1576225p12T10+1799p15T11+p18T12 1 + 1799 T + 1576225 T^{2} + 1115416106 T^{3} + 732038218331 T^{4} + 402383977564751 T^{5} + 190135753613171014 T^{6} + 402383977564751 p^{3} T^{7} + 732038218331 p^{6} T^{8} + 1115416106 p^{9} T^{9} + 1576225 p^{12} T^{10} + 1799 p^{15} T^{11} + p^{18} T^{12}
61 1−625T−246217T2+136351344T3+94023700837T4−21668489332199T5−17403170402691818T6−21668489332199p3T7+94023700837p6T8+136351344p9T9−246217p12T10−625p15T11+p18T12 1 - 625 T - 246217 T^{2} + 136351344 T^{3} + 94023700837 T^{4} - 21668489332199 T^{5} - 17403170402691818 T^{6} - 21668489332199 p^{3} T^{7} + 94023700837 p^{6} T^{8} + 136351344 p^{9} T^{9} - 246217 p^{12} T^{10} - 625 p^{15} T^{11} + p^{18} T^{12}
67 1+495T−430551T2−36746478T3+161353409283T4−31007237744241T5−70581306208381018T6−31007237744241p3T7+161353409283p6T8−36746478p9T9−430551p12T10+495p15T11+p18T12 1 + 495 T - 430551 T^{2} - 36746478 T^{3} + 161353409283 T^{4} - 31007237744241 T^{5} - 70581306208381018 T^{6} - 31007237744241 p^{3} T^{7} + 161353409283 p^{6} T^{8} - 36746478 p^{9} T^{9} - 430551 p^{12} T^{10} + 495 p^{15} T^{11} + p^{18} T^{12}
71 (1−640T+1196037T2−465417472T3+1196037p3T4−640p6T5+p9T6)2 ( 1 - 640 T + 1196037 T^{2} - 465417472 T^{3} + 1196037 p^{3} T^{4} - 640 p^{6} T^{5} + p^{9} T^{6} )^{2}
73 1−443T−406693T2+622914612T3−68154601487T4−127540200423001T5+144654083731272358T6−127540200423001p3T7−68154601487p6T8+622914612p9T9−406693p12T10−443p15T11+p18T12 1 - 443 T - 406693 T^{2} + 622914612 T^{3} - 68154601487 T^{4} - 127540200423001 T^{5} + 144654083731272358 T^{6} - 127540200423001 p^{3} T^{7} - 68154601487 p^{6} T^{8} + 622914612 p^{9} T^{9} - 406693 p^{12} T^{10} - 443 p^{15} T^{11} + p^{18} T^{12}
79 1+pT−940435T2+213350022T3+434920484023T4−120129257254237T5−198873325035594506T6−120129257254237p3T7+434920484023p6T8+213350022p9T9−940435p12T10+p16T11+p18T12 1 + p T - 940435 T^{2} + 213350022 T^{3} + 434920484023 T^{4} - 120129257254237 T^{5} - 198873325035594506 T^{6} - 120129257254237 p^{3} T^{7} + 434920484023 p^{6} T^{8} + 213350022 p^{9} T^{9} - 940435 p^{12} T^{10} + p^{16} T^{11} + p^{18} T^{12}
83 (1−2372T+3356817T2−2979344792T3+3356817p3T4−2372p6T5+p9T6)2 ( 1 - 2372 T + 3356817 T^{2} - 2979344792 T^{3} + 3356817 p^{3} T^{4} - 2372 p^{6} T^{5} + p^{9} T^{6} )^{2}
89 1+821T−895221T2−833200796T3+485399771137T4+256153722115943T5−245058388113291418T6+256153722115943p3T7+485399771137p6T8−833200796p9T9−895221p12T10+821p15T11+p18T12 1 + 821 T - 895221 T^{2} - 833200796 T^{3} + 485399771137 T^{4} + 256153722115943 T^{5} - 245058388113291418 T^{6} + 256153722115943 p^{3} T^{7} + 485399771137 p^{6} T^{8} - 833200796 p^{9} T^{9} - 895221 p^{12} T^{10} + 821 p^{15} T^{11} + p^{18} T^{12}
97 (1+342T+11679T2+841589780T3+11679p3T4+342p6T5+p9T6)2 ( 1 + 342 T + 11679 T^{2} + 841589780 T^{3} + 11679 p^{3} T^{4} + 342 p^{6} T^{5} + p^{9} T^{6} )^{2}
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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

−5.67477483600321017438985428283, −5.18631218294110848349046503401, −5.01242876688093585355879539575, −4.93248694890728635556496067775, −4.82677311773140415896619401721, −4.80702705644141529297265376286, −4.72133954525533282089212281347, −4.19996289360554350656073417885, −4.05034970713377870312402110295, −3.86345278712985361287010084113, −3.66030146026013192656131967534, −3.47328753435172540415278059969, −3.27291564245473603064669454160, −3.18356488266125512763472656157, −2.89895318812325700315437532636, −2.65139779361766959150216945248, −2.12885061169133961700729543221, −2.00894443011009932089532117341, −1.80522746020129402725252227013, −1.62987688819571003712722124716, −1.29676463747758517786720818996, −0.865024692494759317481763741351, −0.78865202048150009922617986408, −0.56819401759378883903723423821, −0.21958077998283339074240728515, 0.21958077998283339074240728515, 0.56819401759378883903723423821, 0.78865202048150009922617986408, 0.865024692494759317481763741351, 1.29676463747758517786720818996, 1.62987688819571003712722124716, 1.80522746020129402725252227013, 2.00894443011009932089532117341, 2.12885061169133961700729543221, 2.65139779361766959150216945248, 2.89895318812325700315437532636, 3.18356488266125512763472656157, 3.27291564245473603064669454160, 3.47328753435172540415278059969, 3.66030146026013192656131967534, 3.86345278712985361287010084113, 4.05034970713377870312402110295, 4.19996289360554350656073417885, 4.72133954525533282089212281347, 4.80702705644141529297265376286, 4.82677311773140415896619401721, 4.93248694890728635556496067775, 5.01242876688093585355879539575, 5.18631218294110848349046503401, 5.67477483600321017438985428283

Graph of the ZZ-function along the critical line

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