Properties

Label 2-15680-1.1-c1-0-44
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  − 3·3-s − 5-s + 6·9-s − 2·11-s + 3·15-s + 4·17-s + 6·19-s − 3·23-s + 25-s − 9·27-s − 9·29-s − 4·31-s + 6·33-s + 4·37-s + 7·41-s − 5·43-s − 6·45-s + 8·47-s − 12·51-s + 2·53-s + 2·55-s − 18·57-s − 10·59-s + 61-s − 9·67-s + 9·69-s − 2·71-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s + 2·9-s − 0.603·11-s + 0.774·15-s + 0.970·17-s + 1.37·19-s − 0.625·23-s + 1/5·25-s − 1.73·27-s − 1.67·29-s − 0.718·31-s + 1.04·33-s + 0.657·37-s + 1.09·41-s − 0.762·43-s − 0.894·45-s + 1.16·47-s − 1.68·51-s + 0.274·53-s + 0.269·55-s − 2.38·57-s − 1.30·59-s + 0.128·61-s − 1.09·67-s + 1.08·69-s − 0.237·71-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 \( 1 + p T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−16.35756374804870, −15.91989117013931, −15.37271933012156, −14.73206445540179, −14.02264489448200, −13.28914958851082, −12.75794653702247, −12.18746128865039, −11.81743285361037, −11.21530771658113, −10.82521928190803, −10.17419908746398, −9.647306116318139, −9.014049304773466, −7.810548886772089, −7.594638361559851, −7.020814676706483, −6.086362568380353, −5.616048835470788, −5.274401723895856, −4.448265412767008, −3.794923880786280, −2.954914040569050, −1.726378232471508, −0.8719549038760187, 0, 0.8719549038760187, 1.726378232471508, 2.954914040569050, 3.794923880786280, 4.448265412767008, 5.274401723895856, 5.616048835470788, 6.086362568380353, 7.020814676706483, 7.594638361559851, 7.810548886772089, 9.014049304773466, 9.647306116318139, 10.17419908746398, 10.82521928190803, 11.21530771658113, 11.81743285361037, 12.18746128865039, 12.75794653702247, 13.28914958851082, 14.02264489448200, 14.73206445540179, 15.37271933012156, 15.91989117013931, 16.35756374804870

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