Properties

Label 2-15680-1.1-c1-0-20
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  + 5-s − 3·9-s − 2·13-s − 2·17-s + 8·19-s + 8·23-s + 25-s − 6·29-s + 2·37-s + 6·41-s − 8·43-s − 3·45-s + 8·47-s + 2·53-s + 8·59-s − 2·61-s − 2·65-s − 8·67-s − 10·73-s − 16·79-s + 9·81-s + 16·83-s − 2·85-s − 10·89-s + 8·95-s + 14·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 9-s − 0.554·13-s − 0.485·17-s + 1.83·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s + 0.328·37-s + 0.937·41-s − 1.21·43-s − 0.447·45-s + 1.16·47-s + 0.274·53-s + 1.04·59-s − 0.256·61-s − 0.248·65-s − 0.977·67-s − 1.17·73-s − 1.80·79-s + 81-s + 1.75·83-s − 0.216·85-s − 1.05·89-s + 0.820·95-s + 1.42·97-s + 0.0995·101-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.028394086\)
\(L(\frac12)\) \(\approx\) \(2.028394086\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−16.07454872562768, −15.37495624423736, −14.71832157293250, −14.48384833808787, −13.68870355372905, −13.31457452661716, −12.76364024210033, −11.89878793395456, −11.55794761211032, −11.00281846315610, −10.33983877748685, −9.651365153864231, −9.066720848023466, −8.807600632538271, −7.783559092213748, −7.342820090844085, −6.708985428270152, −5.847206416502694, −5.378938997680072, −4.886674922766744, −3.907759322705510, −2.999125151405839, −2.658061460331654, −1.587585674329236, −0.6226994877106028, 0.6226994877106028, 1.587585674329236, 2.658061460331654, 2.999125151405839, 3.907759322705510, 4.886674922766744, 5.378938997680072, 5.847206416502694, 6.708985428270152, 7.342820090844085, 7.783559092213748, 8.807600632538271, 9.066720848023466, 9.651365153864231, 10.33983877748685, 11.00281846315610, 11.55794761211032, 11.89878793395456, 12.76364024210033, 13.31457452661716, 13.68870355372905, 14.48384833808787, 14.71832157293250, 15.37495624423736, 16.07454872562768

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