Properties

Label 2-15680-1.1-c1-0-60
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  − 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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
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

−16.47661748907868, −15.91380893327615, −15.07123910644505, −14.66590558812905, −14.18540564702629, −13.15949368944519, −12.89053464905921, −12.52638663250220, −11.53011513000875, −11.33176472600285, −10.68921107269811, −10.15443569281819, −9.591939736400483, −8.943666034976095, −8.156272596479345, −7.438612726626965, −6.998474577323323, −6.096676060313861, −5.686139255906629, −5.078583012879917, −4.739215163608480, −3.468009740023631, −2.939341324981207, −1.894319887200019, −0.9807938942043003, 0, 0.9807938942043003, 1.894319887200019, 2.939341324981207, 3.468009740023631, 4.739215163608480, 5.078583012879917, 5.686139255906629, 6.096676060313861, 6.998474577323323, 7.438612726626965, 8.156272596479345, 8.943666034976095, 9.591939736400483, 10.15443569281819, 10.68921107269811, 11.33176472600285, 11.53011513000875, 12.52638663250220, 12.89053464905921, 13.15949368944519, 14.18540564702629, 14.66590558812905, 15.07123910644505, 15.91380893327615, 16.47661748907868

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