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Properties

Label 2-637-49.39-c1-0-11
Degree $2$
Conductor $637$
Sign $-0.352 - 0.935i$
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  + (0.893 + 0.134i)2-s + (1.14 − 0.782i)3-s + (−1.13 − 0.349i)4-s + (0.316 + 4.22i)5-s + (1.13 − 0.544i)6-s + (−2.63 − 0.243i)7-s + (−2.59 − 1.24i)8-s + (−0.390 + 0.994i)9-s + (−0.285 + 3.81i)10-s + (0.753 + 1.91i)11-s + (−1.57 + 0.485i)12-s + (−0.623 + 0.781i)13-s + (−2.31 − 0.572i)14-s + (3.66 + 4.60i)15-s + (−0.188 − 0.128i)16-s + (−3.82 − 3.55i)17-s + ⋯
L(s)  = 1  + (0.631 + 0.0951i)2-s + (0.662 − 0.451i)3-s + (−0.565 − 0.174i)4-s + (0.141 + 1.88i)5-s + (0.461 − 0.222i)6-s + (−0.995 − 0.0920i)7-s + (−0.916 − 0.441i)8-s + (−0.130 + 0.331i)9-s + (−0.0904 + 1.20i)10-s + (0.227 + 0.578i)11-s + (−0.454 + 0.140i)12-s + (−0.172 + 0.216i)13-s + (−0.620 − 0.152i)14-s + (0.947 + 1.18i)15-s + (−0.0472 − 0.0322i)16-s + (−0.928 − 0.861i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.352 - 0.935i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.352 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.824817 + 1.19252i\)
\(L(\frac12)\) \(\approx\) \(0.824817 + 1.19252i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.63 + 0.243i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
good2 \( 1 + (-0.893 - 0.134i)T + (1.91 + 0.589i)T^{2} \)
3 \( 1 + (-1.14 + 0.782i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (-0.316 - 4.22i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (-0.753 - 1.91i)T + (-8.06 + 7.48i)T^{2} \)
17 \( 1 + (3.82 + 3.55i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (-3.24 - 5.62i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.00756 + 0.00701i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (1.28 - 5.64i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-4.56 + 7.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.37 + 0.731i)T + (30.5 - 20.8i)T^{2} \)
41 \( 1 + (-2.45 - 1.18i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-2.44 + 1.17i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (1.39 + 0.210i)T + (44.9 + 13.8i)T^{2} \)
53 \( 1 + (6.50 + 2.00i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (0.315 - 4.21i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-11.5 + 3.57i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-5.71 + 9.89i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.48 - 10.9i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-2.16 + 0.326i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (-6.68 - 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.35 + 10.4i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (0.898 - 2.29i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−10.79640626219327491988345772182, −9.768670607879758433573180298344, −9.451087162275838795608344428810, −8.006003316965823902008886454757, −7.06928991079337286490578153054, −6.52597883975671056187118372718, −5.53024967959369393625837770318, −4.03447048640776764325129198310, −3.15026932168156192375556362214, −2.35391990262944743518454862252, 0.58709272281626020565590139644, 2.78758436838947790099660059911, 3.84830541057513357631324337330, 4.54949785898378291792440086147, 5.49230452188530297640231435042, 6.39698912606542947318510827709, 8.196149733580337680902660621668, 8.752181869153619000684072234829, 9.287836940725297863459520153759, 9.832496868492085606741227236844

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