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Properties

Label 2-637-91.4-c1-0-6
Degree $2$
Conductor $637$
Sign $0.123 - 0.992i$
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  + 1.38i·2-s + (−1.41 − 2.44i)3-s + 0.0791·4-s + (−0.449 + 0.259i)5-s + (3.39 − 1.95i)6-s + 2.88i·8-s + (−2.49 + 4.31i)9-s + (−0.359 − 0.622i)10-s + (−1.40 + 0.812i)11-s + (−0.111 − 0.193i)12-s + (−1.42 + 3.31i)13-s + (1.26 + 0.733i)15-s − 3.83·16-s + 1.94·17-s + (−5.98 − 3.45i)18-s + (2.15 + 1.24i)19-s + ⋯
L(s)  = 1  + 0.980i·2-s + (−0.815 − 1.41i)3-s + 0.0395·4-s + (−0.200 + 0.116i)5-s + (1.38 − 0.799i)6-s + 1.01i·8-s + (−0.830 + 1.43i)9-s + (−0.113 − 0.196i)10-s + (−0.424 + 0.244i)11-s + (−0.0322 − 0.0559i)12-s + (−0.395 + 0.918i)13-s + (0.327 + 0.189i)15-s − 0.958·16-s + 0.472·17-s + (−1.41 − 0.814i)18-s + (0.494 + 0.285i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.123 - 0.992i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.123 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.777484 + 0.686721i\)
\(L(\frac12)\) \(\approx\) \(0.777484 + 0.686721i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 - 3.31i)T \)
good2 \( 1 - 1.38iT - 2T^{2} \)
3 \( 1 + (1.41 + 2.44i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.449 - 0.259i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.40 - 0.812i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 1.94T + 17T^{2} \)
19 \( 1 + (-2.15 - 1.24i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 9.14T + 23T^{2} \)
29 \( 1 + (-2.61 + 4.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.01 - 2.89i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 10.2iT - 37T^{2} \)
41 \( 1 + (3.64 + 2.10i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.91 - 2.25i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.44 + 7.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.20iT - 59T^{2} \)
61 \( 1 + (6.73 - 11.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.25 + 4.18i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.50 - 2.59i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-10.2 - 5.91i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.491 + 0.850i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.91iT - 83T^{2} \)
89 \( 1 + 12.0iT - 89T^{2} \)
97 \( 1 + (3.82 - 2.21i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.15116867335201152579752772094, −9.925616980344240788358817943939, −8.559434298117743612180366799928, −7.74917324399371998975897073969, −7.03051901001148914335659400988, −6.63539492420666973484061820978, −5.60247999824476807853617389683, −4.86147627927550320893575784324, −2.77665333545400345021126672287, −1.43285594743895069642754447168, 0.66204342675718708175873877878, 2.80291935404990862010264346672, 3.60091496474223699909729248288, 4.75105802168567543788787072760, 5.44980446892137054513097324696, 6.62637238340358817309657240475, 7.85189931763735372795333576811, 9.174895924373871245387593593865, 9.837285252906188225605408716396, 10.65352814840131170501639447447

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