Properties

Label 2-2951-2951.2950-c0-0-13
Degree $2$
Conductor $2951$
Sign $1$
Analytic cond. $1.47274$
Root an. cond. $1.21356$
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 − 3-s + 2·5-s + 6-s + 8-s − 2·10-s + 13-s − 2·15-s − 16-s + 2·23-s − 24-s + 3·25-s − 26-s + 27-s − 29-s + 2·30-s − 31-s − 37-s − 39-s + 2·40-s + 2·41-s − 43-s − 2·46-s + 48-s + 49-s − 3·50-s − 53-s + ⋯
L(s)  = 1  − 2-s − 3-s + 2·5-s + 6-s + 8-s − 2·10-s + 13-s − 2·15-s − 16-s + 2·23-s − 24-s + 3·25-s − 26-s + 27-s − 29-s + 2·30-s − 31-s − 37-s − 39-s + 2·40-s + 2·41-s − 43-s − 2·46-s + 48-s + 49-s − 3·50-s − 53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2951 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2951 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2951\)    =    \(13 \cdot 227\)
Sign: $1$
Analytic conductor: \(1.47274\)
Root analytic conductor: \(1.21356\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2951} (2950, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2951,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7341993900\)
\(L(\frac12)\) \(\approx\) \(0.7341993900\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 - T \)
227 \( 1 - T \)
good2 \( 1 + T + T^{2} \)
3 \( 1 + T + T^{2} \)
5 \( ( 1 - T )^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )^{2} \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T + T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
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

−8.998722315659665088102721563607, −8.627402632786457169198701196965, −7.28092457067035536714753616831, −6.66735535413251197541713026313, −5.75836327678480094397482426569, −5.43984868730231247179106050557, −4.56716977290286835837249510686, −3.05314256108426164839856550010, −1.80913148141158574994936764010, −1.03761234547967850412541522775, 1.03761234547967850412541522775, 1.80913148141158574994936764010, 3.05314256108426164839856550010, 4.56716977290286835837249510686, 5.43984868730231247179106050557, 5.75836327678480094397482426569, 6.66735535413251197541713026313, 7.28092457067035536714753616831, 8.627402632786457169198701196965, 8.998722315659665088102721563607

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