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Properties

Label 2-4056-1.1-c1-0-19
Degree 22
Conductor 40564056
Sign 11
Analytic cond. 32.387332.3873
Root an. cond. 5.690985.69098
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2.71·5-s + 0.735·7-s + 9-s + 1.14·11-s − 2.71·15-s + 5.96·17-s − 3.07·19-s + 0.735·21-s + 5.31·23-s + 2.39·25-s + 27-s − 8.12·29-s − 1.67·31-s + 1.14·33-s − 2.00·35-s − 1.90·37-s + 8.69·41-s + 11.7·43-s − 2.71·45-s − 7.53·47-s − 6.45·49-s + 5.96·51-s + 1.86·53-s − 3.10·55-s − 3.07·57-s − 4.28·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.21·5-s + 0.278·7-s + 0.333·9-s + 0.344·11-s − 0.702·15-s + 1.44·17-s − 0.705·19-s + 0.160·21-s + 1.10·23-s + 0.478·25-s + 0.192·27-s − 1.50·29-s − 0.299·31-s + 0.199·33-s − 0.338·35-s − 0.313·37-s + 1.35·41-s + 1.78·43-s − 0.405·45-s − 1.09·47-s − 0.922·49-s + 0.835·51-s + 0.255·53-s − 0.419·55-s − 0.407·57-s − 0.557·59-s + ⋯

Functional equation

Λ(s)=(4056s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4056s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40564056    =    23⋅3⋅1322^{3} \cdot 3 \cdot 13^{2}
Sign: 11
Analytic conductor: 32.387332.3873
Root analytic conductor: 5.690985.69098
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4056, ( :1/2), 1)(2,\ 4056,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9567736461.956773646
L(12)L(\frac12) ≈\approx 1.9567736461.956773646
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}
ppFp(T)F_p(T)
bad2 1 1
3 1−T 1 - T
13 1 1
good5 1+2.71T+5T2 1 + 2.71T + 5T^{2}
7 1−0.735T+7T2 1 - 0.735T + 7T^{2}
11 1−1.14T+11T2 1 - 1.14T + 11T^{2}
17 1−5.96T+17T2 1 - 5.96T + 17T^{2}
19 1+3.07T+19T2 1 + 3.07T + 19T^{2}
23 1−5.31T+23T2 1 - 5.31T + 23T^{2}
29 1+8.12T+29T2 1 + 8.12T + 29T^{2}
31 1+1.67T+31T2 1 + 1.67T + 31T^{2}
37 1+1.90T+37T2 1 + 1.90T + 37T^{2}
41 1−8.69T+41T2 1 - 8.69T + 41T^{2}
43 1−11.7T+43T2 1 - 11.7T + 43T^{2}
47 1+7.53T+47T2 1 + 7.53T + 47T^{2}
53 1−1.86T+53T2 1 - 1.86T + 53T^{2}
59 1+4.28T+59T2 1 + 4.28T + 59T^{2}
61 1+2.91T+61T2 1 + 2.91T + 61T^{2}
67 1+0.596T+67T2 1 + 0.596T + 67T^{2}
71 1−2.30T+71T2 1 - 2.30T + 71T^{2}
73 1−13.1T+73T2 1 - 13.1T + 73T^{2}
79 1−12.6T+79T2 1 - 12.6T + 79T^{2}
83 1−9.10T+83T2 1 - 9.10T + 83T^{2}
89 1−8.77T+89T2 1 - 8.77T + 89T^{2}
97 1+6.86T+97T2 1 + 6.86T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.264311817543293700092118026135, −7.70738194191150754913516085148, −7.32507016216675852961706205000, −6.32147117028371872348662408590, −5.36102469624808980829449377019, −4.48796299814743107151708971411, −3.72743283637119550623362156495, −3.19831522720164251665746945645, −1.99192125140785734051341695848, −0.789997458263512926483671216196, 0.789997458263512926483671216196, 1.99192125140785734051341695848, 3.19831522720164251665746945645, 3.72743283637119550623362156495, 4.48796299814743107151708971411, 5.36102469624808980829449377019, 6.32147117028371872348662408590, 7.32507016216675852961706205000, 7.70738194191150754913516085148, 8.264311817543293700092118026135

Graph of the ZZ-function along the critical line