Properties

Label 2-15680-1.1-c1-0-13
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  − 2·3-s + 5-s + 9-s − 4·11-s + 2·13-s − 2·15-s − 8·17-s + 6·19-s + 4·23-s + 25-s + 4·27-s + 6·29-s + 4·31-s + 8·33-s + 10·37-s − 4·39-s − 4·41-s + 4·43-s + 45-s + 4·47-s + 16·51-s − 10·53-s − 4·55-s − 12·57-s + 14·59-s − 10·61-s + 2·65-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 − 1.94·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 1.39·33-s + 1.64·37-s − 0.640·39-s − 0.624·41-s + 0.609·43-s + 0.149·45-s + 0.583·47-s + 2.24·51-s − 1.37·53-s − 0.539·55-s − 1.58·57-s + 1.82·59-s − 1.28·61-s + 0.248·65-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

\(L(1)\) \(\approx\) \(1.149305988\)
\(L(\frac12)\) \(\approx\) \(1.149305988\)
\(L(\frac{3}{2})\) 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 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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

−15.99719291991362, −15.68049615145653, −15.05895300017780, −14.21967899282248, −13.58306595799075, −13.21475099356458, −12.73608089327235, −11.88967252471231, −11.49424431117243, −10.86946547683726, −10.58171934759825, −9.878124781195322, −9.179402469053965, −8.577724687907205, −7.903126969476521, −7.096805581771087, −6.537413117170917, −5.970495520802647, −5.406747203847850, −4.770947594515083, −4.317866390306817, −2.940061582860227, −2.624554573963832, −1.374340604744287, −0.5247999545666088, 0.5247999545666088, 1.374340604744287, 2.624554573963832, 2.940061582860227, 4.317866390306817, 4.770947594515083, 5.406747203847850, 5.970495520802647, 6.537413117170917, 7.096805581771087, 7.903126969476521, 8.577724687907205, 9.179402469053965, 9.878124781195322, 10.58171934759825, 10.86946547683726, 11.49424431117243, 11.88967252471231, 12.73608089327235, 13.21475099356458, 13.58306595799075, 14.21967899282248, 15.05895300017780, 15.68049615145653, 15.99719291991362

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