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Properties

Label 2-234-1.1-c1-0-3
Degree 22
Conductor 234234
Sign 11
Analytic cond. 1.868491.86849
Root an. cond. 1.366931.36693
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-s + 4-s + 3·5-s − 7-s + 8-s + 3·10-s − 6·11-s + 13-s − 14-s + 16-s + 3·17-s + 2·19-s + 3·20-s − 6·22-s + 4·25-s + 26-s − 28-s − 6·29-s − 4·31-s + 32-s + 3·34-s − 3·35-s − 7·37-s + 2·38-s + 3·40-s − 43-s − 6·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s + 0.353·8-s + 0.948·10-s − 1.80·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.458·19-s + 0.670·20-s − 1.27·22-s + 4/5·25-s + 0.196·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s − 0.507·35-s − 1.15·37-s + 0.324·38-s + 0.474·40-s − 0.152·43-s − 0.904·44-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 11
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), 1)(2,\ 234,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0087998962.008799896
L(12)L(\frac12) ≈\approx 2.0087998962.008799896
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−T 1 - T
3 1 1
13 1−T 1 - T
good5 1−3T+pT2 1 - 3 T + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1+6T+pT2 1 + 6 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1+3T+pT2 1 + 3 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1−14T+pT2 1 - 14 T + p T^{2}
71 1−3T+pT2 1 - 3 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+10T+pT2 1 + 10 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

−12.57399662018781984180868180363, −11.16559240112924374874969825279, −10.25096846899164676242515061975, −9.569105792671075621984918394603, −8.109212031498814193717580943346, −6.94019412385488865960361148937, −5.66417880764699758598698858020, −5.24164848449732929783448594193, −3.34467457020301574582723277384, −2.09859100780083292577684604843, 2.09859100780083292577684604843, 3.34467457020301574582723277384, 5.24164848449732929783448594193, 5.66417880764699758598698858020, 6.94019412385488865960361148937, 8.109212031498814193717580943346, 9.569105792671075621984918394603, 10.25096846899164676242515061975, 11.16559240112924374874969825279, 12.57399662018781984180868180363

Graph of the ZZ-function along the critical line