Properties

Label 2-15680-1.1-c1-0-38
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  + 2·3-s + 5-s + 9-s + 3·11-s − 13-s + 2·15-s + 6·17-s + 19-s − 9·23-s + 25-s − 4·27-s − 6·29-s + 8·31-s + 6·33-s + 7·37-s − 2·39-s − 3·41-s + 2·43-s + 45-s + 9·47-s + 12·51-s − 9·53-s + 3·55-s + 2·57-s + 8·61-s − 65-s + 8·67-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 0.516·15-s + 1.45·17-s + 0.229·19-s − 1.87·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 1.43·31-s + 1.04·33-s + 1.15·37-s − 0.320·39-s − 0.468·41-s + 0.304·43-s + 0.149·45-s + 1.31·47-s + 1.68·51-s − 1.23·53-s + 0.404·55-s + 0.264·57-s + 1.02·61-s − 0.124·65-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

\(L(1)\) \(\approx\) \(4.096956906\)
\(L(\frac12)\) \(\approx\) \(4.096956906\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.83570269136988, −15.39362572251935, −14.60003442162991, −14.28867703712750, −14.03085778100858, −13.38148402334137, −12.73914718884628, −11.99850356327809, −11.73944015715067, −10.85551808198842, −9.998641033846153, −9.679721654032897, −9.281799253515570, −8.468139015182436, −7.951393511655794, −7.569066786200730, −6.667022018559333, −5.965802312401329, −5.505575657893073, −4.464209869427197, −3.795488489001087, −3.255612710976934, −2.410282460072281, −1.823365038722067, −0.8354684100824752, 0.8354684100824752, 1.823365038722067, 2.410282460072281, 3.255612710976934, 3.795488489001087, 4.464209869427197, 5.505575657893073, 5.965802312401329, 6.667022018559333, 7.569066786200730, 7.951393511655794, 8.468139015182436, 9.281799253515570, 9.679721654032897, 9.998641033846153, 10.85551808198842, 11.73944015715067, 11.99850356327809, 12.73914718884628, 13.38148402334137, 14.03085778100858, 14.28867703712750, 14.60003442162991, 15.39362572251935, 15.83570269136988

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