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Properties

Label 4-612e2-1.1-c1e2-0-4
Degree 44
Conductor 374544374544
Sign 11
Analytic cond. 23.881223.8812
Root an. cond. 2.210622.21062
Motivic weight 11
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  − 2·5-s − 4·7-s + 8·11-s + 8·17-s + 12·23-s + 2·25-s − 10·29-s + 12·31-s + 8·35-s − 6·37-s + 2·41-s − 8·47-s + 8·49-s − 16·55-s − 14·61-s + 8·67-s + 4·71-s + 22·73-s − 32·77-s + 20·79-s − 16·85-s + 12·89-s + 2·97-s − 20·101-s − 16·103-s + 8·107-s − 2·109-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s + 2.41·11-s + 1.94·17-s + 2.50·23-s + 2/5·25-s − 1.85·29-s + 2.15·31-s + 1.35·35-s − 0.986·37-s + 0.312·41-s − 1.16·47-s + 8/7·49-s − 2.15·55-s − 1.79·61-s + 0.977·67-s + 0.474·71-s + 2.57·73-s − 3.64·77-s + 2.25·79-s − 1.73·85-s + 1.27·89-s + 0.203·97-s − 1.99·101-s − 1.57·103-s + 0.773·107-s − 0.191·109-s + ⋯

Functional equation

Λ(s)=(374544s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 374544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(374544s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 374544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 374544374544    =    24⋅34⋅1722^{4} \cdot 3^{4} \cdot 17^{2}
Sign: 11
Analytic conductor: 23.881223.8812
Root analytic conductor: 2.210622.21062
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 374544, ( :1/2,1/2), 1)(4,\ 374544,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6424787171.642478717
L(12)L(\frac12) ≈\approx 1.6424787171.642478717
L(32)L(\frac{3}{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)
bad2 1 1
3 1 1
17C2C_2 1−8T+pT2 1 - 8 T + p T^{2}
good5C2C_2 (1−2T+pT2)(1+4T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )
7C22C_2^2 1+4T+8T2+4pT3+p2T4 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−8T+32T2−8pT3+p2T4 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
23C22C_2^2 1−12T+72T2−12pT3+p2T4 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+10T+50T2+10pT3+p2T4 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4}
31C22C_2^2 1−12T+72T2−12pT3+p2T4 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+6T+18T2+6pT3+p2T4 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4}
41C2C_2 (1−10T+pT2)(1+8T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} )
43C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
47C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
61C22C_2^2 1+14T+98T2+14pT3+p2T4 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4}
67C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
71C22C_2^2 1−4T+8T2−4pT3+p2T4 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4}
73C2C_2 (1−16T+pT2)(1−6T+pT2) ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} )
79C22C_2^2 1−20T+200T2−20pT3+p2T4 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−150T2+p2T4 1 - 150 T^{2} + p^{2} T^{4}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C22C_2^2 1−2T+2T2−2pT3+p2T4 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} 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

−10.73832354680183839005206663074, −10.57666994473032512311495035219, −9.712943332079620702904224880717, −9.399053246256000894384537368025, −9.374925459912937453641427683254, −8.854669748143288446263038339853, −8.091462973279646554223775897170, −7.86357144834143598832965002568, −7.17029040357322757241518491399, −6.72803835872275106961877198295, −6.53430347233623719086091638147, −6.09576564322952022389165580841, −5.21231917020507877968658723982, −4.96466095759736728144939614278, −3.90805101118437922388867547692, −3.75370463950568375824643881756, −3.30778476311261671731741503051, −2.78581035484535109265896794328, −1.42468603163083608634659134336, −0.814687359582873198965572374691, 0.814687359582873198965572374691, 1.42468603163083608634659134336, 2.78581035484535109265896794328, 3.30778476311261671731741503051, 3.75370463950568375824643881756, 3.90805101118437922388867547692, 4.96466095759736728144939614278, 5.21231917020507877968658723982, 6.09576564322952022389165580841, 6.53430347233623719086091638147, 6.72803835872275106961877198295, 7.17029040357322757241518491399, 7.86357144834143598832965002568, 8.091462973279646554223775897170, 8.854669748143288446263038339853, 9.374925459912937453641427683254, 9.399053246256000894384537368025, 9.712943332079620702904224880717, 10.57666994473032512311495035219, 10.73832354680183839005206663074

Graph of the ZZ-function along the critical line