Properties

Label 2-15680-1.1-c1-0-39
Degree $2$
Conductor $15680$
Sign $1$
Analytic cond. $125.205$
Root an. cond. $11.1895$
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  + 3·3-s + 5-s + 6·9-s − 5·11-s − 5·13-s + 3·15-s + 7·17-s + 2·19-s + 2·23-s + 25-s + 9·27-s − 7·29-s + 4·31-s − 15·33-s + 6·37-s − 15·39-s + 12·41-s − 2·43-s + 6·45-s + 47-s + 21·51-s − 5·55-s + 6·57-s + 4·59-s + 4·61-s − 5·65-s + 8·67-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s + 2·9-s − 1.50·11-s − 1.38·13-s + 0.774·15-s + 1.69·17-s + 0.458·19-s + 0.417·23-s + 1/5·25-s + 1.73·27-s − 1.29·29-s + 0.718·31-s − 2.61·33-s + 0.986·37-s − 2.40·39-s + 1.87·41-s − 0.304·43-s + 0.894·45-s + 0.145·47-s + 2.94·51-s − 0.674·55-s + 0.794·57-s + 0.520·59-s + 0.512·61-s − 0.620·65-s + 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15680\)    =    \(2^{6} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(125.205\)
Root analytic conductor: \(11.1895\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{15680} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.620603158\)
\(L(\frac12)\) \(\approx\) \(4.620603158\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good3 \( 1 - p T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−15.92569025719452, −15.13373486288676, −14.87754304979238, −14.32019758386306, −13.93048416453304, −13.23083440908293, −12.80938037750294, −12.42855178637713, −11.53371487328102, −10.66633071577337, −10.05026621923427, −9.626105589332534, −9.360696339588295, −8.395950110455690, −7.892888934450544, −7.509245684830943, −7.084403983715135, −5.849658710998730, −5.307515950576888, −4.625848780164993, −3.750772724612488, −2.935424483826749, −2.630030531608772, −1.949236212584872, −0.8399407915507089, 0.8399407915507089, 1.949236212584872, 2.630030531608772, 2.935424483826749, 3.750772724612488, 4.625848780164993, 5.307515950576888, 5.849658710998730, 7.084403983715135, 7.509245684830943, 7.892888934450544, 8.395950110455690, 9.360696339588295, 9.626105589332534, 10.05026621923427, 10.66633071577337, 11.53371487328102, 12.42855178637713, 12.80938037750294, 13.23083440908293, 13.93048416453304, 14.32019758386306, 14.87754304979238, 15.13373486288676, 15.92569025719452

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