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Properties

Label 2-637-13.12-c1-0-8
Degree $2$
Conductor $637$
Sign $-0.798 - 0.601i$
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  + 1.17i·2-s − 0.539·3-s + 0.630·4-s + 0.460i·5-s − 0.630i·6-s + 3.07i·8-s − 2.70·9-s − 0.539·10-s + 0.829i·11-s − 0.340·12-s + (2.87 + 2.17i)13-s − 0.248i·15-s − 2.34·16-s − 2.87·17-s − 3.17i·18-s + 4.32i·19-s + ⋯
L(s)  = 1  + 0.827i·2-s − 0.311·3-s + 0.315·4-s + 0.206i·5-s − 0.257i·6-s + 1.08i·8-s − 0.903·9-s − 0.170·10-s + 0.250i·11-s − 0.0981·12-s + (0.798 + 0.601i)13-s − 0.0641i·15-s − 0.585·16-s − 0.698·17-s − 0.747i·18-s + 0.992i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.798 - 0.601i$
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: \((2,\ 637,\ (\ :1/2),\ -0.798 - 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.398769 + 1.19164i\)
\(L(\frac12)\) \(\approx\) \(0.398769 + 1.19164i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.87 - 2.17i)T \)
good2 \( 1 - 1.17iT - 2T^{2} \)
3 \( 1 + 0.539T + 3T^{2} \)
5 \( 1 - 0.460iT - 5T^{2} \)
11 \( 1 - 0.829iT - 11T^{2} \)
17 \( 1 + 2.87T + 17T^{2} \)
19 \( 1 - 4.32iT - 19T^{2} \)
23 \( 1 + 5.04T + 23T^{2} \)
29 \( 1 - 0.261T + 29T^{2} \)
31 \( 1 - 6.80iT - 31T^{2} \)
37 \( 1 - 9.51iT - 37T^{2} \)
41 \( 1 + 6.68iT - 41T^{2} \)
43 \( 1 - 0.418T + 43T^{2} \)
47 \( 1 + 9.24iT - 47T^{2} \)
53 \( 1 - 1.63T + 53T^{2} \)
59 \( 1 - 2.78iT - 59T^{2} \)
61 \( 1 + 7.26T + 61T^{2} \)
67 \( 1 - 5.07iT - 67T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + 0.353iT - 73T^{2} \)
79 \( 1 - 2.81T + 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + 5.43iT - 89T^{2} \)
97 \( 1 + 12.6iT - 97T^{2} \)
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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

−10.94002754834782043975658230207, −10.22585995085905129592107950489, −8.767637953180453463181827082519, −8.358728578792698498993847798982, −7.18364919890574372608888726007, −6.42213997788841037874159753426, −5.79624890033987750436802415283, −4.73437413379863539094759331377, −3.29020351358403761912655909648, −1.92332482781661870489225873009, 0.68473158128734591095679135972, 2.31406729583108467397504705361, 3.27817609460783243359897119902, 4.46799073174058105414722910058, 5.80759499743016437480369061414, 6.44199581941371605024625130689, 7.65157725706393817687760140173, 8.674555477797384266032338489371, 9.473458573001836950808647091356, 10.61810802232941733993443076982

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