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Properties

Label 2-1323-1.1-c1-0-10
Degree 22
Conductor 13231323
Sign 11
Analytic cond. 10.564210.5642
Root an. cond. 3.250263.25026
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 5·13-s + 4·16-s + 7·19-s − 5·25-s + 4·31-s + 11·37-s + 8·43-s + 10·52-s + 61-s − 8·64-s + 5·67-s + 7·73-s − 14·76-s + 17·79-s + 19·97-s + 10·100-s + 13·103-s + 2·109-s + ⋯
L(s)  = 1  − 4-s − 1.38·13-s + 16-s + 1.60·19-s − 25-s + 0.718·31-s + 1.80·37-s + 1.21·43-s + 1.38·52-s + 0.128·61-s − 64-s + 0.610·67-s + 0.819·73-s − 1.60·76-s + 1.91·79-s + 1.92·97-s + 100-s + 1.28·103-s + 0.191·109-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 11
Analytic conductor: 10.564210.5642
Root analytic conductor: 3.250263.25026
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1323, ( :1/2), 1)(2,\ 1323,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1565365421.156536542
L(12)L(\frac12) ≈\approx 1.1565365421.156536542
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)
bad3 1 1
7 1 1
good2 1+pT2 1 + p T^{2}
5 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−7T+pT2 1 - 7 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−11T+pT2 1 - 11 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−T+pT2 1 - T + p T^{2}
67 1−5T+pT2 1 - 5 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1−17T+pT2 1 - 17 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−19T+pT2 1 - 19 T + p T^{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

−9.692549886033731115950097335871, −9.003116301093652454576632877528, −7.83187954508823562149152602221, −7.53309190588908388391290020645, −6.19032093295709689179707806358, −5.27555176123571387068189863753, −4.61317128196167712173450637604, −3.61858250543880863713873154871, −2.48005247853672969845362220771, −0.794908582887598964605828839390, 0.794908582887598964605828839390, 2.48005247853672969845362220771, 3.61858250543880863713873154871, 4.61317128196167712173450637604, 5.27555176123571387068189863753, 6.19032093295709689179707806358, 7.53309190588908388391290020645, 7.83187954508823562149152602221, 9.003116301093652454576632877528, 9.692549886033731115950097335871

Graph of the ZZ-function along the critical line