Properties

Label 2-15680-1.1-c1-0-16
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 2·9-s − 6·11-s + 2·13-s − 15-s + 6·17-s + 8·19-s + 3·23-s + 25-s − 5·27-s − 3·29-s − 2·31-s − 6·33-s − 8·37-s + 2·39-s + 3·41-s − 5·43-s + 2·45-s + 6·51-s − 12·53-s + 6·55-s + 8·57-s − 61-s − 2·65-s + 7·67-s + 3·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 2/3·9-s − 1.80·11-s + 0.554·13-s − 0.258·15-s + 1.45·17-s + 1.83·19-s + 0.625·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s − 0.359·31-s − 1.04·33-s − 1.31·37-s + 0.320·39-s + 0.468·41-s − 0.762·43-s + 0.298·45-s + 0.840·51-s − 1.64·53-s + 0.809·55-s + 1.05·57-s − 0.128·61-s − 0.248·65-s + 0.855·67-s + 0.361·69-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

\(L(1)\) \(\approx\) \(1.816795573\)
\(L(\frac12)\) \(\approx\) \(1.816795573\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−15.90889178326795, −15.52290697580212, −14.89259234832691, −14.23605694963095, −13.88766712368576, −13.25339642521316, −12.73830650796264, −12.07132543621424, −11.49492200705564, −10.91719268016665, −10.37797362981413, −9.645890656819274, −9.186551340721934, −8.346688800670895, −7.878472316419055, −7.610896209518821, −6.839377679207846, −5.718304933565735, −5.392083740367081, −4.865763029819278, −3.524794229377930, −3.316446484380378, −2.690608758997877, −1.646000676638222, −0.5605304428189968, 0.5605304428189968, 1.646000676638222, 2.690608758997877, 3.316446484380378, 3.524794229377930, 4.865763029819278, 5.392083740367081, 5.718304933565735, 6.839377679207846, 7.610896209518821, 7.878472316419055, 8.346688800670895, 9.186551340721934, 9.645890656819274, 10.37797362981413, 10.91719268016665, 11.49492200705564, 12.07132543621424, 12.73830650796264, 13.25339642521316, 13.88766712368576, 14.23605694963095, 14.89259234832691, 15.52290697580212, 15.90889178326795

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