Properties

Label 2-637-1.1-c1-0-15
Degree $2$
Conductor $637$
Sign $1$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·4-s + 3·5-s + 9-s − 4·12-s − 13-s + 6·15-s + 4·16-s + 6·17-s + 7·19-s − 6·20-s + 3·23-s + 4·25-s − 4·27-s − 9·29-s − 5·31-s − 2·36-s + 2·37-s − 2·39-s + 6·41-s − 43-s + 3·45-s − 3·47-s + 8·48-s + 12·51-s + 2·52-s − 9·53-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s + 1.34·5-s + 1/3·9-s − 1.15·12-s − 0.277·13-s + 1.54·15-s + 16-s + 1.45·17-s + 1.60·19-s − 1.34·20-s + 0.625·23-s + 4/5·25-s − 0.769·27-s − 1.67·29-s − 0.898·31-s − 1/3·36-s + 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.152·43-s + 0.447·45-s − 0.437·47-s + 1.15·48-s + 1.68·51-s + 0.277·52-s − 1.23·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{637} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.192809362\)
\(L(\frac12)\) \(\approx\) \(2.192809362\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + T \)
good2 \( 1 + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−10.03148147866603089067459950733, −9.513361358049510508727228598963, −9.166717133010691319523246651397, −8.059674511475772558890445124278, −7.35100084789034730201905858533, −5.69003044725444814418386644498, −5.29343454176495501876748811750, −3.73616092552496566051685235363, −2.86956267148131998918823072126, −1.46947183400268699699754331247, 1.46947183400268699699754331247, 2.86956267148131998918823072126, 3.73616092552496566051685235363, 5.29343454176495501876748811750, 5.69003044725444814418386644498, 7.35100084789034730201905858533, 8.059674511475772558890445124278, 9.166717133010691319523246651397, 9.513361358049510508727228598963, 10.03148147866603089067459950733

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