Properties

Label 2-15680-1.1-c1-0-52
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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 2·9-s − 3·11-s + 5·13-s + 15-s − 3·17-s − 2·19-s + 6·23-s + 25-s + 5·27-s − 3·29-s − 4·31-s + 3·33-s − 2·37-s − 5·39-s + 12·41-s − 10·43-s + 2·45-s + 9·47-s + 3·51-s − 12·53-s + 3·55-s + 2·57-s + 8·61-s − 5·65-s − 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.904·11-s + 1.38·13-s + 0.258·15-s − 0.727·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s − 0.718·31-s + 0.522·33-s − 0.328·37-s − 0.800·39-s + 1.87·41-s − 1.52·43-s + 0.298·45-s + 1.31·47-s + 0.420·51-s − 1.64·53-s + 0.404·55-s + 0.264·57-s + 1.02·61-s − 0.620·65-s − 0.488·67-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 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.27918058683733, −15.70521205619242, −15.34866029952413, −14.63538749084514, −14.08799270286763, −13.31053556615522, −12.94368303078648, −12.46612160133975, −11.50298562059002, −11.23528110807813, −10.80675915703285, −10.33180779386039, −9.261472568973643, −8.802161270444983, −8.318985739712541, −7.594941507796575, −6.944351412433016, −6.220785031076169, −5.738494598494589, −5.055805374194028, −4.414894259348672, −3.559915844925244, −2.943367486130850, −2.040492009437095, −0.9144680225949532, 0, 0.9144680225949532, 2.040492009437095, 2.943367486130850, 3.559915844925244, 4.414894259348672, 5.055805374194028, 5.738494598494589, 6.220785031076169, 6.944351412433016, 7.594941507796575, 8.318985739712541, 8.802161270444983, 9.261472568973643, 10.33180779386039, 10.80675915703285, 11.23528110807813, 11.50298562059002, 12.46612160133975, 12.94368303078648, 13.31053556615522, 14.08799270286763, 14.63538749084514, 15.34866029952413, 15.70521205619242, 16.27918058683733

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