Properties

Label 2-15680-1.1-c1-0-23
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 + 4·11-s + 2·13-s − 2·15-s − 2·19-s + 4·23-s + 25-s + 4·27-s − 10·29-s + 4·31-s − 8·33-s + 2·37-s − 4·39-s + 12·41-s − 4·43-s + 45-s + 4·47-s − 2·53-s + 4·55-s + 4·57-s − 10·59-s + 6·61-s + 2·65-s + 4·67-s − 8·69-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s − 0.458·19-s + 0.834·23-s + 1/5·25-s + 0.769·27-s − 1.85·29-s + 0.718·31-s − 1.39·33-s + 0.328·37-s − 0.640·39-s + 1.87·41-s − 0.609·43-s + 0.149·45-s + 0.583·47-s − 0.274·53-s + 0.539·55-s + 0.529·57-s − 1.30·59-s + 0.768·61-s + 0.248·65-s + 0.488·67-s − 0.963·69-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

\(L(1)\) \(\approx\) \(1.663645753\)
\(L(\frac12)\) \(\approx\) \(1.663645753\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−16.11648899136381, −15.53819216791032, −14.84524388208862, −14.33724069126200, −13.82759694289416, −12.98246020126257, −12.70332085274632, −11.98719987704537, −11.34370632607985, −11.09563448010884, −10.55131705046124, −9.694390529637441, −9.236758228375639, −8.688642995857843, −7.868252370906429, −7.035524807330864, −6.523352357891857, −5.975253678527251, −5.549634177328490, −4.738603035968676, −4.100231993600531, −3.338284628966599, −2.318061357481807, −1.382554821971984, −0.6510173931768425, 0.6510173931768425, 1.382554821971984, 2.318061357481807, 3.338284628966599, 4.100231993600531, 4.738603035968676, 5.549634177328490, 5.975253678527251, 6.523352357891857, 7.035524807330864, 7.868252370906429, 8.688642995857843, 9.236758228375639, 9.694390529637441, 10.55131705046124, 11.09563448010884, 11.34370632607985, 11.98719987704537, 12.70332085274632, 12.98246020126257, 13.82759694289416, 14.33724069126200, 14.84524388208862, 15.53819216791032, 16.11648899136381

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