↑

Properties

Label 2-910-1.1-c1-0-24
Degree $2$
Conductor $910$
Sign $-1$
Analytic cond. $7.26638$
Root an. cond. $2.69562$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s − 7-s + 8-s − 3·9-s − 10-s − 6·11-s + 13-s − 14-s + 16-s − 8·17-s − 3·18-s − 20-s − 6·22-s + 25-s + 26-s − 28-s + 6·29-s − 2·31-s + 32-s − 8·34-s + 35-s − 3·36-s + 10·37-s − 40-s − 8·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 9-s − 0.316·10-s − 1.80·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.94·17-s − 0.707·18-s − 0.223·20-s − 1.27·22-s + 1/5·25-s + 0.196·26-s − 0.188·28-s + 1.11·29-s − 0.359·31-s + 0.176·32-s − 1.37·34-s + 0.169·35-s − 1/2·36-s + 1.64·37-s − 0.158·40-s − 1.24·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(910\)    =    \(2 \cdot 5 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(7.26638\)
Root analytic conductor: \(2.69562\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 910,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 - T \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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

−9.800771313969527415908990405125, −8.588702015015171454309716376061, −8.063034858535104995372368196768, −6.96001729612816463304152118858, −6.13314547658354194000697316834, −5.19002518169606038376181212548, −4.38040480288368631324962363899, −3.09520116202287492831446930127, −2.40510846377760824680251914161, 0, 2.40510846377760824680251914161, 3.09520116202287492831446930127, 4.38040480288368631324962363899, 5.19002518169606038376181212548, 6.13314547658354194000697316834, 6.96001729612816463304152118858, 8.063034858535104995372368196768, 8.588702015015171454309716376061, 9.800771313969527415908990405125

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