Properties

Label 2-637-49.37-c1-0-8
Degree $2$
Conductor $637$
Sign $-0.888 - 0.458i$
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  + (−1.69 + 1.15i)2-s + (1.28 − 1.19i)3-s + (0.811 − 2.06i)4-s + (0.653 + 0.201i)5-s + (−0.799 + 3.50i)6-s + (−2.34 + 1.21i)7-s + (0.100 + 0.441i)8-s + (0.00460 − 0.0614i)9-s + (−1.34 + 0.414i)10-s + (0.294 + 3.92i)11-s + (−1.41 − 3.61i)12-s + (−0.900 − 0.433i)13-s + (2.58 − 4.78i)14-s + (1.07 − 0.519i)15-s + (2.57 + 2.38i)16-s + (−5.35 − 0.807i)17-s + ⋯
L(s)  = 1  + (−1.20 + 0.818i)2-s + (0.740 − 0.687i)3-s + (0.405 − 1.03i)4-s + (0.292 + 0.0902i)5-s + (−0.326 + 1.43i)6-s + (−0.888 + 0.459i)7-s + (0.0355 + 0.155i)8-s + (0.00153 − 0.0204i)9-s + (−0.424 + 0.131i)10-s + (0.0887 + 1.18i)11-s + (−0.409 − 1.04i)12-s + (−0.249 − 0.120i)13-s + (0.689 − 1.27i)14-s + (0.278 − 0.134i)15-s + (0.643 + 0.597i)16-s + (−1.29 − 0.195i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.888 - 0.458i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123815 + 0.509705i\)
\(L(\frac12)\) \(\approx\) \(0.123815 + 0.509705i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.34 - 1.21i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
good2 \( 1 + (1.69 - 1.15i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (-1.28 + 1.19i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (-0.653 - 0.201i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (-0.294 - 3.92i)T + (-10.8 + 1.63i)T^{2} \)
17 \( 1 + (5.35 + 0.807i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (0.297 - 0.515i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.45 - 0.520i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (-1.81 + 2.27i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-0.831 - 1.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.23 - 8.24i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-2.08 - 9.15i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.0187 + 0.0821i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (8.34 - 5.68i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (4.69 - 11.9i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-1.08 + 0.333i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (4.00 + 10.1i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-1.57 - 2.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.70 - 5.89i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (12.8 + 8.77i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (-3.08 + 5.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.64 + 2.23i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-0.572 + 7.63i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 6.03T + 97T^{2} \)
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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

−10.42351078689932749340138627961, −9.676744129907982907794447508792, −9.189341673854298450961110830303, −8.176704129641653623397040291548, −7.65389304370113339723840610363, −6.64292339161488991076122744966, −6.22728830061646356140707598430, −4.56688313583767420189450385844, −2.84565684196327935039661009439, −1.78693548698538647369138098574, 0.36420783366255801434205782811, 2.18103807817236985640986863552, 3.24519183687027887727572732753, 4.07855192384505732898647061328, 5.74430404688264917725250321383, 6.84054726695070266705452288707, 8.117256866984844460919674003000, 8.849121395906467870884746394597, 9.382241177694389367436395157898, 10.02668303681967293101489991515

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