Properties

Label 2-85176-1.1-c1-0-35
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  + 2·5-s + 7-s + 3·17-s − 2·19-s + 23-s − 25-s + 10·29-s + 5·31-s + 2·35-s − 2·37-s + 10·41-s − 43-s + 10·47-s + 49-s + 9·53-s − 9·59-s − 61-s − 9·67-s − 9·71-s − 4·73-s + 4·79-s − 83-s + 6·85-s + 15·89-s − 4·95-s + 12·97-s + 101-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s + 0.727·17-s − 0.458·19-s + 0.208·23-s − 1/5·25-s + 1.85·29-s + 0.898·31-s + 0.338·35-s − 0.328·37-s + 1.56·41-s − 0.152·43-s + 1.45·47-s + 1/7·49-s + 1.23·53-s − 1.17·59-s − 0.128·61-s − 1.09·67-s − 1.06·71-s − 0.468·73-s + 0.450·79-s − 0.109·83-s + 0.650·85-s + 1.58·89-s − 0.410·95-s + 1.21·97-s + 0.0995·101-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: $\chi_{85176} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 85176,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.937951512\)
\(L(\frac12)\) \(\approx\) \(3.937951512\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−13.88693000595803, −13.60045777142129, −12.95698593303307, −12.46269494045153, −11.83248080232722, −11.70782384740102, −10.66453792192194, −10.44449134160729, −10.10985451725105, −9.350759329250063, −8.953607344330063, −8.476203114273882, −7.746586475959991, −7.455727888144967, −6.638431996179078, −6.124507179588076, −5.795365424586064, −5.101055124855759, −4.545635837354125, −4.060039006847075, −3.149302387195118, −2.623623615565554, −2.046735997669852, −1.257852402845001, −0.6890556932952180, 0.6890556932952180, 1.257852402845001, 2.046735997669852, 2.623623615565554, 3.149302387195118, 4.060039006847075, 4.545635837354125, 5.101055124855759, 5.795365424586064, 6.124507179588076, 6.638431996179078, 7.455727888144967, 7.746586475959991, 8.476203114273882, 8.953607344330063, 9.350759329250063, 10.10985451725105, 10.44449134160729, 10.66453792192194, 11.70782384740102, 11.83248080232722, 12.46269494045153, 12.95698593303307, 13.60045777142129, 13.88693000595803

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