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Properties

Label 2-2695-55.54-c0-0-9
Degree $2$
Conductor $2695$
Sign $1$
Analytic cond. $1.34498$
Root an. cond. $1.15973$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 5-s + 8-s + 9-s − 10-s + 11-s − 13-s − 16-s + 2·17-s − 18-s − 22-s + 25-s + 26-s − 31-s − 2·34-s + 40-s − 43-s + 45-s − 50-s + 55-s − 59-s + 62-s + 64-s − 65-s − 71-s + 72-s − 73-s + ⋯
L(s)  = 1  − 2-s + 5-s + 8-s + 9-s − 10-s + 11-s − 13-s − 16-s + 2·17-s − 18-s − 22-s + 25-s + 26-s − 31-s − 2·34-s + 40-s − 43-s + 45-s − 50-s + 55-s − 59-s + 62-s + 64-s − 65-s − 71-s + 72-s − 73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2695\)    =    \(5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(1.34498\)
Root analytic conductor: \(1.15973\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2695} (1814, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2695,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9493221856\)
\(L(\frac12)\) \(\approx\) \(0.9493221856\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + T + T^{2} \)
3 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T + T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.221794053437689668690348303361, −8.428522512928050774621782742616, −7.42708686755809798317485345367, −7.11535625591917661627564229810, −6.02285511588967042612794442161, −5.15796786223111297483692753034, −4.37202446885125778043441482152, −3.27044774257802213984610453279, −1.84293378050175807563833964232, −1.20717714747313553103586371497, 1.20717714747313553103586371497, 1.84293378050175807563833964232, 3.27044774257802213984610453279, 4.37202446885125778043441482152, 5.15796786223111297483692753034, 6.02285511588967042612794442161, 7.11535625591917661627564229810, 7.42708686755809798317485345367, 8.428522512928050774621782742616, 9.221794053437689668690348303361

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