Properties

Label 2-15680-1.1-c1-0-40
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 − 11-s − 3·13-s − 3·15-s − 3·17-s − 6·19-s − 4·23-s + 25-s − 9·27-s + 29-s + 6·31-s + 3·33-s + 9·39-s + 6·41-s + 6·43-s + 6·45-s − 9·47-s + 9·51-s + 10·53-s − 55-s + 18·57-s + 6·59-s − 3·65-s + 14·67-s + 12·69-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s + 2·9-s − 0.301·11-s − 0.832·13-s − 0.774·15-s − 0.727·17-s − 1.37·19-s − 0.834·23-s + 1/5·25-s − 1.73·27-s + 0.185·29-s + 1.07·31-s + 0.522·33-s + 1.44·39-s + 0.937·41-s + 0.914·43-s + 0.894·45-s − 1.31·47-s + 1.26·51-s + 1.37·53-s − 0.134·55-s + 2.38·57-s + 0.781·59-s − 0.372·65-s + 1.71·67-s + 1.44·69-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: \(1\)
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 \( 1 + T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 15 T + p T^{2} \)
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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

−16.34597150818796, −15.88237434301519, −15.32847239879278, −14.67963337058737, −14.02527607274658, −13.26407624945671, −12.75209402853527, −12.40685942332952, −11.70010928197103, −11.26159180304908, −10.66733668918765, −10.11271575313026, −9.820214470817458, −8.887946965594850, −8.192039013172773, −7.387802858599121, −6.720167889931798, −6.293642099924552, −5.745791152250997, −5.067438592249797, −4.514321039088832, −3.976293979182003, −2.575651535825041, −1.994159049888275, −0.8475101528438349, 0, 0.8475101528438349, 1.994159049888275, 2.575651535825041, 3.976293979182003, 4.514321039088832, 5.067438592249797, 5.745791152250997, 6.293642099924552, 6.720167889931798, 7.387802858599121, 8.192039013172773, 8.887946965594850, 9.820214470817458, 10.11271575313026, 10.66733668918765, 11.26159180304908, 11.70010928197103, 12.40685942332952, 12.75209402853527, 13.26407624945671, 14.02527607274658, 14.67963337058737, 15.32847239879278, 15.88237434301519, 16.34597150818796

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