Properties

Label 2-85176-1.1-c1-0-4
Degree $2$
Conductor $85176$
Sign $1$
Analytic cond. $680.133$
Root an. cond. $26.0793$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 7-s − 5·11-s − 7·17-s − 3·19-s − 3·23-s + 4·25-s − 7·29-s + 4·31-s + 3·35-s − 11·37-s − 4·41-s + 43-s − 8·47-s + 49-s − 2·53-s − 15·55-s − 6·59-s + 13·61-s − 4·67-s + 6·71-s − 73-s − 5·77-s + 8·79-s − 14·83-s − 21·85-s − 9·95-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s − 1.50·11-s − 1.69·17-s − 0.688·19-s − 0.625·23-s + 4/5·25-s − 1.29·29-s + 0.718·31-s + 0.507·35-s − 1.80·37-s − 0.624·41-s + 0.152·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s − 2.02·55-s − 0.781·59-s + 1.66·61-s − 0.488·67-s + 0.712·71-s − 0.117·73-s − 0.569·77-s + 0.900·79-s − 1.53·83-s − 2.27·85-s − 0.923·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85176\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(680.133\)
Root analytic conductor: \(26.0793\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 85176,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.063755514\)
\(L(\frac12)\) \(\approx\) \(1.063755514\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
13 \( 1 \)
good5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.84279654482526, −13.40412887025869, −13.03996943509311, −12.70097452740961, −11.96581457273029, −11.27824575326138, −10.90028958758394, −10.41053756387510, −9.980934277415854, −9.545474836045102, −8.749582149700619, −8.563100356935313, −7.904818438593899, −7.277552656469586, −6.643584822038961, −6.249787008824148, −5.587226628092580, −5.151611064594917, −4.708307002696286, −3.989295431877868, −3.184326696248007, −2.410178734076865, −2.042032573892057, −1.619426707091632, −0.2983621205161657, 0.2983621205161657, 1.619426707091632, 2.042032573892057, 2.410178734076865, 3.184326696248007, 3.989295431877868, 4.708307002696286, 5.151611064594917, 5.587226628092580, 6.249787008824148, 6.643584822038961, 7.277552656469586, 7.904818438593899, 8.563100356935313, 8.749582149700619, 9.545474836045102, 9.980934277415854, 10.41053756387510, 10.90028958758394, 11.27824575326138, 11.96581457273029, 12.70097452740961, 13.03996943509311, 13.40412887025869, 13.84279654482526

Graph of the $Z$-function along the critical line