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Properties

Label 4-315e2-1.1-c3e2-0-2
Degree 44
Conductor 9922599225
Sign 11
Analytic cond. 345.424345.424
Root an. cond. 4.311104.31110
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 34·4-s + 10·5-s − 14·7-s − 96·8-s − 80·10-s + 14·11-s + 50·13-s + 112·14-s + 196·16-s + 50·17-s + 36·19-s + 340·20-s − 112·22-s − 244·23-s + 75·25-s − 400·26-s − 476·28-s + 26·29-s − 120·31-s − 352·32-s − 400·34-s − 140·35-s + 564·37-s − 288·38-s − 960·40-s + 328·41-s + ⋯
L(s)  = 1  − 2.82·2-s + 17/4·4-s + 0.894·5-s − 0.755·7-s − 4.24·8-s − 2.52·10-s + 0.383·11-s + 1.06·13-s + 2.13·14-s + 3.06·16-s + 0.713·17-s + 0.434·19-s + 3.80·20-s − 1.08·22-s − 2.21·23-s + 3/5·25-s − 3.01·26-s − 3.21·28-s + 0.166·29-s − 0.695·31-s − 1.94·32-s − 2.01·34-s − 0.676·35-s + 2.50·37-s − 1.22·38-s − 3.79·40-s + 1.24·41-s + ⋯

Functional equation

Λ(s)=(99225s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(99225s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9922599225    =    34⋅52⋅723^{4} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 345.424345.424
Root analytic conductor: 4.311104.31110
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 99225, ( :3/2,3/2), 1)(4,\ 99225,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.79523175780.7952317578
L(12)L(\frac12) ≈\approx 0.79523175780.7952317578
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
5C1C_1 (1−pT)2 ( 1 - p T )^{2}
7C1C_1 (1+pT)2 ( 1 + p T )^{2}
good2D4D_{4} 1+p3T+15pT2+p6T3+p6T4 1 + p^{3} T + 15 p T^{2} + p^{6} T^{3} + p^{6} T^{4}
11D4D_{4} 1−14T+663T2−14p3T3+p6T4 1 - 14 T + 663 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−50T+4987T2−50p3T3+p6T4 1 - 50 T + 4987 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−50T+387pT2−50p3T3+p6T4 1 - 50 T + 387 p T^{2} - 50 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−36T+10170T2−36p3T3+p6T4 1 - 36 T + 10170 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+244T+29970T2+244p3T3+p6T4 1 + 244 T + 29970 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−26T+47795T2−26p3T3+p6T4 1 - 26 T + 47795 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+120T−1618T2+120p3T3+p6T4 1 + 120 T - 1618 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−564T+173630T2−564p3T3+p6T4 1 - 564 T + 173630 T^{2} - 564 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−8pT+133986T2−8p4T3+p6T4 1 - 8 p T + 133986 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4}
43D4D_{4} 1+260T+166666T2+260p3T3+p6T4 1 + 260 T + 166666 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−350T+203423T2−350p3T3+p6T4 1 - 350 T + 203423 T^{2} - 350 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−56T+265770T2−56p3T3+p6T4 1 - 56 T + 265770 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4}
59C2C_2 (1−616T+p3T2)2 ( 1 - 616 T + p^{3} T^{2} )^{2}
61D4D_{4} 1−336T+458858T2−336p3T3+p6T4 1 - 336 T + 458858 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+152T+599110T2+152p3T3+p6T4 1 + 152 T + 599110 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4}
71C2C_2 (1−952T+p3T2)2 ( 1 - 952 T + p^{3} T^{2} )^{2}
73D4D_{4} 1−676T+655606T2−676p3T3+p6T4 1 - 676 T + 655606 T^{2} - 676 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−1014T+1120119T2−1014p3T3+p6T4 1 - 1014 T + 1120119 T^{2} - 1014 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−376T+458918T2−376p3T3+p6T4 1 - 376 T + 458918 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−216T+1417730T2−216p3T3+p6T4 1 - 216 T + 1417730 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−2742T+3608187T2−2742p3T3+p6T4 1 - 2742 T + 3608187 T^{2} - 2742 p^{3} T^{3} + p^{6} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.05045810834759882923289655509, −10.82595620169439374386044625293, −10.03506708339436406126858493542, −9.991102482284658848926355405589, −9.589197403687613012345096762524, −9.256755215362388152699174337459, −8.765080844569788964574115615407, −8.250511204401108437830195045069, −7.85796112771084375656852881012, −7.51468412112048442240412634178, −6.67226643040837680714918820400, −6.39462802812522710275969801258, −5.85166306504775962032271663219, −5.33990507519871392978892684932, −4.00330066673777330629276434001, −3.55385585016490645879943977057, −2.35611731956218467381565749202, −2.02065331865541195418603221681, −0.878670259546278060408011684066, −0.75815921534852244250859828078, 0.75815921534852244250859828078, 0.878670259546278060408011684066, 2.02065331865541195418603221681, 2.35611731956218467381565749202, 3.55385585016490645879943977057, 4.00330066673777330629276434001, 5.33990507519871392978892684932, 5.85166306504775962032271663219, 6.39462802812522710275969801258, 6.67226643040837680714918820400, 7.51468412112048442240412634178, 7.85796112771084375656852881012, 8.250511204401108437830195045069, 8.765080844569788964574115615407, 9.256755215362388152699174337459, 9.589197403687613012345096762524, 9.991102482284658848926355405589, 10.03506708339436406126858493542, 10.82595620169439374386044625293, 11.05045810834759882923289655509

Graph of the ZZ-function along the critical line