Properties

Label 2-15680-1.1-c1-0-3
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-s − 5-s − 2·9-s + 11-s − 5·13-s − 15-s − 17-s − 6·19-s − 4·23-s + 25-s − 5·27-s − 3·29-s − 2·31-s + 33-s − 8·37-s − 5·39-s + 10·41-s + 2·43-s + 2·45-s + 7·47-s − 51-s + 2·53-s − 55-s − 6·57-s + 14·59-s − 8·61-s + 5·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.301·11-s − 1.38·13-s − 0.258·15-s − 0.242·17-s − 1.37·19-s − 0.834·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s − 0.359·31-s + 0.174·33-s − 1.31·37-s − 0.800·39-s + 1.56·41-s + 0.304·43-s + 0.298·45-s + 1.02·47-s − 0.140·51-s + 0.274·53-s − 0.134·55-s − 0.794·57-s + 1.82·59-s − 1.02·61-s + 0.620·65-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

\(L(1)\) \(\approx\) \(0.9789152127\)
\(L(\frac12)\) \(\approx\) \(0.9789152127\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−15.96643006420942, −15.22996792961332, −14.87847851411285, −14.38178009191936, −13.96710958752622, −13.22473105864816, −12.55688988234670, −12.16052482085672, −11.56396837252248, −10.89800763515341, −10.37858649331164, −9.629911248478251, −9.039614406689464, −8.595580472826370, −7.943598478379674, −7.367651621776429, −6.829343542218148, −5.923530770231000, −5.435504668515462, −4.407704843217932, −4.065504814962749, −3.167546016769368, −2.415157973868554, −1.901113066004515, −0.3855327755231226, 0.3855327755231226, 1.901113066004515, 2.415157973868554, 3.167546016769368, 4.065504814962749, 4.407704843217932, 5.435504668515462, 5.923530770231000, 6.829343542218148, 7.367651621776429, 7.943598478379674, 8.595580472826370, 9.039614406689464, 9.629911248478251, 10.37858649331164, 10.89800763515341, 11.56396837252248, 12.16052482085672, 12.55688988234670, 13.22473105864816, 13.96710958752622, 14.38178009191936, 14.87847851411285, 15.22996792961332, 15.96643006420942

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