Properties

Label 2-15680-1.1-c1-0-54
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 2·9-s − 5·11-s + 7·13-s + 15-s + 3·17-s + 2·19-s − 8·23-s + 25-s + 5·27-s + 5·29-s − 10·31-s + 5·33-s − 4·37-s − 7·39-s + 6·41-s + 2·43-s + 2·45-s − 7·47-s − 3·51-s + 10·53-s + 5·55-s − 2·57-s + 10·59-s − 12·61-s − 7·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s − 1.50·11-s + 1.94·13-s + 0.258·15-s + 0.727·17-s + 0.458·19-s − 1.66·23-s + 1/5·25-s + 0.962·27-s + 0.928·29-s − 1.79·31-s + 0.870·33-s − 0.657·37-s − 1.12·39-s + 0.937·41-s + 0.304·43-s + 0.298·45-s − 1.02·47-s − 0.420·51-s + 1.37·53-s + 0.674·55-s − 0.264·57-s + 1.30·59-s − 1.53·61-s − 0.868·65-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: \(1\)
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 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−16.18938213166535, −16.02334747809605, −15.30252349102919, −14.59349972217641, −13.96895968062941, −13.53101819612347, −12.90382159142202, −12.25878751969482, −11.79193585528278, −11.15758024057582, −10.65811864610189, −10.35629903615455, −9.448610945767935, −8.675242531901043, −8.158700389795393, −7.795910467959152, −6.952731846180855, −6.131544685822729, −5.637249506246858, −5.290859319214401, −4.280512968634299, −3.566638181009949, −2.996559508995736, −2.004695710442037, −0.9291316241278127, 0, 0.9291316241278127, 2.004695710442037, 2.996559508995736, 3.566638181009949, 4.280512968634299, 5.290859319214401, 5.637249506246858, 6.131544685822729, 6.952731846180855, 7.795910467959152, 8.158700389795393, 8.675242531901043, 9.448610945767935, 10.35629903615455, 10.65811864610189, 11.15758024057582, 11.79193585528278, 12.25878751969482, 12.90382159142202, 13.53101819612347, 13.96895968062941, 14.59349972217641, 15.30252349102919, 16.02334747809605, 16.18938213166535

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