Properties

Label 2-15680-1.1-c1-0-101
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 3·9-s − 4·11-s + 2·13-s − 6·17-s − 4·19-s − 8·23-s + 25-s − 6·29-s − 10·37-s − 10·41-s − 12·43-s + 3·45-s + 6·53-s + 4·55-s + 12·59-s + 2·61-s − 2·65-s − 4·67-s − 12·71-s + 10·73-s − 4·79-s + 9·81-s + 6·85-s − 18·89-s + 4·95-s + 10·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 9-s − 1.20·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s − 1.64·37-s − 1.56·41-s − 1.82·43-s + 0.447·45-s + 0.824·53-s + 0.539·55-s + 1.56·59-s + 0.256·61-s − 0.248·65-s − 0.488·67-s − 1.42·71-s + 1.17·73-s − 0.450·79-s + 81-s + 0.650·85-s − 1.90·89-s + 0.410·95-s + 1.01·97-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: \(2\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−16.49319676722293, −15.94743845539388, −15.37057032054731, −15.08319237643297, −14.34822718514144, −13.57873662227067, −13.37322662599175, −12.72140078515619, −11.92232539024458, −11.53286052336063, −10.95896822247662, −10.37537476684336, −9.932372182094418, −8.815022341085080, −8.502382453469194, −8.184638941297086, −7.261755650747709, −6.682474694309237, −5.967924828572936, −5.367976581714014, −4.714114284389263, −3.860207613351864, −3.331714427842910, −2.328837138424458, −1.858364442440314, 0, 0, 1.858364442440314, 2.328837138424458, 3.331714427842910, 3.860207613351864, 4.714114284389263, 5.367976581714014, 5.967924828572936, 6.682474694309237, 7.261755650747709, 8.184638941297086, 8.502382453469194, 8.815022341085080, 9.932372182094418, 10.37537476684336, 10.95896822247662, 11.53286052336063, 11.92232539024458, 12.72140078515619, 13.37322662599175, 13.57873662227067, 14.34822718514144, 15.08319237643297, 15.37057032054731, 15.94743845539388, 16.49319676722293

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