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Properties

Label 2-637-13.12-c1-0-22
Degree $2$
Conductor $637$
Sign $0.554 + 0.832i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 2·3-s − 2·4-s + i·5-s − 4i·6-s + 9-s − 2·10-s + 2i·11-s + 4·12-s + (−2 − 3i)13-s − 2i·15-s − 4·16-s − 6·17-s + 2i·18-s − 3i·19-s − 2i·20-s + ⋯
L(s)  = 1  + 1.41i·2-s − 1.15·3-s − 4-s + 0.447i·5-s − 1.63i·6-s + 0.333·9-s − 0.632·10-s + 0.603i·11-s + 1.15·12-s + (−0.554 − 0.832i)13-s − 0.516i·15-s − 16-s − 1.45·17-s + 0.471i·18-s − 0.688i·19-s − 0.447i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.554 + 0.832i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.554 + 0.832i)\)

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)$
bad7 \( 1 \)
13 \( 1 + (2 + 3i)T \)
good2 \( 1 - 2iT - 2T^{2} \)
3 \( 1 + 2T + 3T^{2} \)
5 \( 1 - iT - 5T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 3iT - 19T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 3iT - 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + 11iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 - 8iT - 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 14iT - 71T^{2} \)
73 \( 1 - 9iT - 73T^{2} \)
79 \( 1 + 9T + 79T^{2} \)
83 \( 1 - 11iT - 83T^{2} \)
89 \( 1 - 5iT - 89T^{2} \)
97 \( 1 + 9iT - 97T^{2} \)
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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

−10.60188472996887045646898003745, −9.455999512320706268133758776845, −8.435868471268412834282927978456, −7.46845644063795032700415055949, −6.69857726537634178221491136525, −6.15981812264780274989695574912, −5.14506032513818329594458958615, −4.52976834198199688116305868134, −2.55512658368274325442703096465, 0, 1.45588659791679540377998257182, 2.79570865269400158326389904892, 4.21027217027925131061760650220, 4.90435296613595806539306822092, 6.13657304669700031967935774496, 6.91319204113181846728039349163, 8.487830139178467269102932636535, 9.241153728365169954617798729300, 10.22945587824056141253117744610

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