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Properties

Label 2-637-13.12-c1-0-41
Degree $2$
Conductor $637$
Sign $-0.698 - 0.715i$
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  − 2.12i·2-s + 0.178·3-s − 2.51·4-s − 3.60i·5-s − 0.380i·6-s + 1.10i·8-s − 2.96·9-s − 7.66·10-s − 3.99i·11-s − 0.450·12-s + (2.51 + 2.58i)13-s − 0.644i·15-s − 2.69·16-s + 4.78·17-s + 6.30i·18-s + 3.15i·19-s + ⋯
L(s)  = 1  − 1.50i·2-s + 0.103·3-s − 1.25·4-s − 1.61i·5-s − 0.155i·6-s + 0.389i·8-s − 0.989·9-s − 2.42·10-s − 1.20i·11-s − 0.129·12-s + (0.698 + 0.715i)13-s − 0.166i·15-s − 0.674·16-s + 1.16·17-s + 1.48i·18-s + 0.722i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.698 - 0.715i$
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: \((2,\ 637,\ (\ :1/2),\ -0.698 - 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.468571 + 1.11169i\)
\(L(\frac12)\) \(\approx\) \(0.468571 + 1.11169i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.51 - 2.58i)T \)
good2 \( 1 + 2.12iT - 2T^{2} \)
3 \( 1 - 0.178T + 3T^{2} \)
5 \( 1 + 3.60iT - 5T^{2} \)
11 \( 1 + 3.99iT - 11T^{2} \)
17 \( 1 - 4.78T + 17T^{2} \)
19 \( 1 - 3.15iT - 19T^{2} \)
23 \( 1 - 2.17T + 23T^{2} \)
29 \( 1 + 6.57T + 29T^{2} \)
31 \( 1 + 1.48iT - 31T^{2} \)
37 \( 1 + 4.96iT - 37T^{2} \)
41 \( 1 - 2.11iT - 41T^{2} \)
43 \( 1 + 1.43T + 43T^{2} \)
47 \( 1 - 1.01iT - 47T^{2} \)
53 \( 1 - 6.03T + 53T^{2} \)
59 \( 1 + 4.90iT - 59T^{2} \)
61 \( 1 - 2.03T + 61T^{2} \)
67 \( 1 - 3.91iT - 67T^{2} \)
71 \( 1 + 8.80iT - 71T^{2} \)
73 \( 1 - 3.08iT - 73T^{2} \)
79 \( 1 - 1.96T + 79T^{2} \)
83 \( 1 + 7.66iT - 83T^{2} \)
89 \( 1 + 12.7iT - 89T^{2} \)
97 \( 1 + 1.35iT - 97T^{2} \)
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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

−10.09779806052729556022626924254, −9.073423144505634746873068892081, −8.804788225858397274169665130405, −7.86242787773914291176570586064, −6.01038375053300685429503553368, −5.26619174874063714565822879162, −4.00777168712623235426110506854, −3.26050158311004187629663844137, −1.74300689835329088115083321613, −0.66106060975948088744780987542, 2.53388006234002145834242973900, 3.59466217451917091211120100188, 5.16108414559937293485613439039, 5.92812760008804702306996236998, 6.79990703587413853641404565658, 7.41382382559469459126014281074, 8.137801084915949681982114664911, 9.177090443028639721901399517979, 10.18023970578191011517679752679, 11.01238592180417237089980697909

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