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Properties

Label 2-637-13.12-c1-0-6
Degree $2$
Conductor $637$
Sign $-0.983 + 0.181i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.65i·2-s + 0.494·3-s − 0.740·4-s + 2.87i·5-s + 0.819i·6-s + 2.08i·8-s − 2.75·9-s − 4.75·10-s − 2.58i·11-s − 0.366·12-s + (−3.54 + 0.654i)13-s + 1.42i·15-s − 4.93·16-s − 2.27·17-s − 4.56i·18-s + 4.39i·19-s + ⋯
L(s)  = 1  + 1.17i·2-s + 0.285·3-s − 0.370·4-s + 1.28i·5-s + 0.334i·6-s + 0.737i·8-s − 0.918·9-s − 1.50·10-s − 0.778i·11-s − 0.105·12-s + (−0.983 + 0.181i)13-s + 0.367i·15-s − 1.23·16-s − 0.551·17-s − 1.07i·18-s + 1.00i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.983 + 0.181i$
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: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.983 + 0.181i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.119512 - 1.30544i\)
\(L(\frac12)\) \(\approx\) \(0.119512 - 1.30544i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.54 - 0.654i)T \)
good2 \( 1 - 1.65iT - 2T^{2} \)
3 \( 1 - 0.494T + 3T^{2} \)
5 \( 1 - 2.87iT - 5T^{2} \)
11 \( 1 + 2.58iT - 11T^{2} \)
17 \( 1 + 2.27T + 17T^{2} \)
19 \( 1 - 4.39iT - 19T^{2} \)
23 \( 1 - 7.16T + 23T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 - 6.85iT - 31T^{2} \)
37 \( 1 + 5.89iT - 37T^{2} \)
41 \( 1 - 3.98iT - 41T^{2} \)
43 \( 1 - 1.31T + 43T^{2} \)
47 \( 1 - 1.35iT - 47T^{2} \)
53 \( 1 + 8.28T + 53T^{2} \)
59 \( 1 + 9.61iT - 59T^{2} \)
61 \( 1 - 9.55T + 61T^{2} \)
67 \( 1 - 15.9iT - 67T^{2} \)
71 \( 1 - 4.79iT - 71T^{2} \)
73 \( 1 - 5.70iT - 73T^{2} \)
79 \( 1 + 2.01T + 79T^{2} \)
83 \( 1 + 3.11iT - 83T^{2} \)
89 \( 1 + 13.0iT - 89T^{2} \)
97 \( 1 - 14.0iT - 97T^{2} \)
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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

−11.09075708956978934885418511123, −10.18687961499842539423386173287, −8.942157406538943443533689963823, −8.280075501216363641030821235059, −7.32275782086583719299999829776, −6.68788184892765497239992547739, −5.91090017894673872201773714270, −4.93088679327046018407899410338, −3.25349832938073515688366619919, −2.48080360854641910650735168198, 0.65443836029599994622976451237, 2.15756853696236106719346247699, 3.03017556925435061769601212313, 4.50358136461410609177178691860, 5.05075992164591481709446117916, 6.57255200259660565909594930281, 7.64022244593665093084687306195, 8.825489230443594263477252087406, 9.254527361476063210285397158099, 10.11931591768096198523810957512

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