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Properties

Label 6-153e3-1.1-c3e3-0-1
Degree 66
Conductor 35815773581577
Sign −1-1
Analytic cond. 735.652735.652
Root an. cond. 3.004543.00454
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 33

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 7·4-s − 8·5-s − 8·7-s + 9·8-s + 40·10-s − 34·11-s + 36·13-s + 40·14-s − 85·16-s − 51·17-s − 142·19-s − 56·20-s + 170·22-s − 110·23-s − 252·25-s − 180·26-s − 56·28-s − 90·29-s − 148·31-s + 341·32-s + 255·34-s + 64·35-s + 110·37-s + 710·38-s − 72·40-s − 720·41-s + ⋯
L(s)  = 1  − 1.76·2-s + 7/8·4-s − 0.715·5-s − 0.431·7-s + 0.397·8-s + 1.26·10-s − 0.931·11-s + 0.768·13-s + 0.763·14-s − 1.32·16-s − 0.727·17-s − 1.71·19-s − 0.626·20-s + 1.64·22-s − 0.997·23-s − 2.01·25-s − 1.35·26-s − 0.377·28-s − 0.576·29-s − 0.857·31-s + 1.88·32-s + 1.28·34-s + 0.309·35-s + 0.488·37-s + 3.03·38-s − 0.284·40-s − 2.74·41-s + ⋯

Functional equation

Λ(s)=(3581577s/2 ΓC(s)3 L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 3581577 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(3581577s/2 ΓC(s+3/2)3 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3581577 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 66
Conductor: 35815773581577    =    36⋅1733^{6} \cdot 17^{3}
Sign: −1-1
Analytic conductor: 735.652735.652
Root analytic conductor: 3.004543.00454
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 33
Selberg data: (6, 3581577, ( :3/2,3/2,3/2), −1)(6,\ 3581577,\ (\ :3/2, 3/2, 3/2),\ -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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3 1 1
17C1C_1 (1+pT)3 ( 1 + p T )^{3}
good2S4×C2S_4\times C_2 1+5T+9pT2+23pT3+9p4T4+5p6T5+p9T6 1 + 5 T + 9 p T^{2} + 23 p T^{3} + 9 p^{4} T^{4} + 5 p^{6} T^{5} + p^{9} T^{6}
5S4×C2S_4\times C_2 1+8T+316T2+1534T3+316p3T4+8p6T5+p9T6 1 + 8 T + 316 T^{2} + 1534 T^{3} + 316 p^{3} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6}
7S4×C2S_4\times C_2 1+8T+51pT2+7792T3+51p4T4+8p6T5+p9T6 1 + 8 T + 51 p T^{2} + 7792 T^{3} + 51 p^{4} T^{4} + 8 p^{6} T^{5} + p^{9} T^{6}
11S4×C2S_4\times C_2 1+34T+2450T2+99472T3+2450p3T4+34p6T5+p9T6 1 + 34 T + 2450 T^{2} + 99472 T^{3} + 2450 p^{3} T^{4} + 34 p^{6} T^{5} + p^{9} T^{6}
13S4×C2S_4\times C_2 1−36T+1060T2−35486T3+1060p3T4−36p6T5+p9T6 1 - 36 T + 1060 T^{2} - 35486 T^{3} + 1060 p^{3} T^{4} - 36 p^{6} T^{5} + p^{9} T^{6}
19S4×C2S_4\times C_2 1+142T+24690T2+1956200T3+24690p3T4+142p6T5+p9T6 1 + 142 T + 24690 T^{2} + 1956200 T^{3} + 24690 p^{3} T^{4} + 142 p^{6} T^{5} + p^{9} T^{6}
23S4×C2S_4\times C_2 1+110T+30814T2+2729980T3+30814p3T4+110p6T5+p9T6 1 + 110 T + 30814 T^{2} + 2729980 T^{3} + 30814 p^{3} T^{4} + 110 p^{6} T^{5} + p^{9} T^{6}
29S4×C2S_4\times C_2 1+90T+36139T2+4805340T3+36139p3T4+90p6T5+p9T6 1 + 90 T + 36139 T^{2} + 4805340 T^{3} + 36139 p^{3} T^{4} + 90 p^{6} T^{5} + p^{9} T^{6}
31S4×C2S_4\times C_2 1+148T+82401T2+8177688T3+82401p3T4+148p6T5+p9T6 1 + 148 T + 82401 T^{2} + 8177688 T^{3} + 82401 p^{3} T^{4} + 148 p^{6} T^{5} + p^{9} T^{6}
37S4×C2S_4\times C_2 1−110T+71031T2−17113452T3+71031p3T4−110p6T5+p9T6 1 - 110 T + 71031 T^{2} - 17113452 T^{3} + 71031 p^{3} T^{4} - 110 p^{6} T^{5} + p^{9} T^{6}
41S4×C2S_4\times C_2 1+720T+366208T2+109686282T3+366208p3T4+720p6T5+p9T6 1 + 720 T + 366208 T^{2} + 109686282 T^{3} + 366208 p^{3} T^{4} + 720 p^{6} T^{5} + p^{9} T^{6}
43S4×C2S_4\times C_2 1+146T−40182T2−39408872T3−40182p3T4+146p6T5+p9T6 1 + 146 T - 40182 T^{2} - 39408872 T^{3} - 40182 p^{3} T^{4} + 146 p^{6} T^{5} + p^{9} T^{6}
47S4×C2S_4\times C_2 1+500T+217553T2+73350104T3+217553p3T4+500p6T5+p9T6 1 + 500 T + 217553 T^{2} + 73350104 T^{3} + 217553 p^{3} T^{4} + 500 p^{6} T^{5} + p^{9} T^{6}
53S4×C2S_4\times C_2 1+610T+340931T2+101182252T3+340931p3T4+610p6T5+p9T6 1 + 610 T + 340931 T^{2} + 101182252 T^{3} + 340931 p^{3} T^{4} + 610 p^{6} T^{5} + p^{9} T^{6}
59S4×C2S_4\times C_2 1−216T+576349T2−90026112T3+576349p3T4−216p6T5+p9T6 1 - 216 T + 576349 T^{2} - 90026112 T^{3} + 576349 p^{3} T^{4} - 216 p^{6} T^{5} + p^{9} T^{6}
61S4×C2S_4\times C_2 1+18T+539791T2+16298516T3+539791p3T4+18p6T5+p9T6 1 + 18 T + 539791 T^{2} + 16298516 T^{3} + 539791 p^{3} T^{4} + 18 p^{6} T^{5} + p^{9} T^{6}
67S4×C2S_4\times C_2 1+1404T+1477377T2+906611816T3+1477377p3T4+1404p6T5+p9T6 1 + 1404 T + 1477377 T^{2} + 906611816 T^{3} + 1477377 p^{3} T^{4} + 1404 p^{6} T^{5} + p^{9} T^{6}
71S4×C2S_4\times C_2 1−960T+856597T2−459564544T3+856597p3T4−960p6T5+p9T6 1 - 960 T + 856597 T^{2} - 459564544 T^{3} + 856597 p^{3} T^{4} - 960 p^{6} T^{5} + p^{9} T^{6}
73S4×C2S_4\times C_2 1+794T+908231T2+390276652T3+908231p3T4+794p6T5+p9T6 1 + 794 T + 908231 T^{2} + 390276652 T^{3} + 908231 p^{3} T^{4} + 794 p^{6} T^{5} + p^{9} T^{6}
79S4×C2S_4\times C_2 1+276T+332913T2+51343320T3+332913p3T4+276p6T5+p9T6 1 + 276 T + 332913 T^{2} + 51343320 T^{3} + 332913 p^{3} T^{4} + 276 p^{6} T^{5} + p^{9} T^{6}
83S4×C2S_4\times C_2 1−1552T+2256501T2−1786088240T3+2256501p3T4−1552p6T5+p9T6 1 - 1552 T + 2256501 T^{2} - 1786088240 T^{3} + 2256501 p^{3} T^{4} - 1552 p^{6} T^{5} + p^{9} T^{6}
89S4×C2S_4\times C_2 1+1394T+2215963T2+1686994660T3+2215963p3T4+1394p6T5+p9T6 1 + 1394 T + 2215963 T^{2} + 1686994660 T^{3} + 2215963 p^{3} T^{4} + 1394 p^{6} T^{5} + p^{9} T^{6}
97S4×C2S_4\times C_2 1−402T+119587T2+1292278940T3+119587p3T4−402p6T5+p9T6 1 - 402 T + 119587 T^{2} + 1292278940 T^{3} + 119587 p^{3} T^{4} - 402 p^{6} T^{5} + p^{9} T^{6}
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   L(s)=∏p ∏j=16(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.65644661908976076971321866031, −11.21163196919064666561215266294, −10.96965997318068539077218433579, −10.50400401685843674619413223268, −10.16420266429647573781039250093, −9.838401726963965250614617301965, −9.778701373800632889397996087190, −9.068366168463588903802152142660, −8.897996537578487955617633066410, −8.644243741732146534086964003420, −8.142182592671096268013463040669, −7.88229179255241623027027399994, −7.83474739392216022566928054928, −7.22694930000157491336706397957, −6.56917714299354251397802095120, −6.32754042281170777383415357055, −6.17627826640342648198481797484, −5.17720788170549705638842089984, −5.08004569547810011465154599049, −4.14971281380827279224524226351, −4.08902997194734833753093184761, −3.47203228415881083087051326614, −2.77838577401269894070988374506, −1.80764492040638991603309607579, −1.75646561566549658193007977758, 0, 0, 0, 1.75646561566549658193007977758, 1.80764492040638991603309607579, 2.77838577401269894070988374506, 3.47203228415881083087051326614, 4.08902997194734833753093184761, 4.14971281380827279224524226351, 5.08004569547810011465154599049, 5.17720788170549705638842089984, 6.17627826640342648198481797484, 6.32754042281170777383415357055, 6.56917714299354251397802095120, 7.22694930000157491336706397957, 7.83474739392216022566928054928, 7.88229179255241623027027399994, 8.142182592671096268013463040669, 8.644243741732146534086964003420, 8.897996537578487955617633066410, 9.068366168463588903802152142660, 9.778701373800632889397996087190, 9.838401726963965250614617301965, 10.16420266429647573781039250093, 10.50400401685843674619413223268, 10.96965997318068539077218433579, 11.21163196919064666561215266294, 11.65644661908976076971321866031

Graph of the ZZ-function along the critical line