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Properties

Label 2-637-91.74-c1-0-40
Degree $2$
Conductor $637$
Sign $-0.997 - 0.0769i$
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  + 1.73·2-s + (−1.36 − 2.36i)3-s + 0.999·4-s + (−0.866 − 1.5i)5-s + (−2.36 − 4.09i)6-s − 1.73·8-s + (−2.23 + 3.86i)9-s + (−1.49 − 2.59i)10-s + (−0.633 − 1.09i)11-s + (−1.36 − 2.36i)12-s + (3.59 − 0.232i)13-s + (−2.36 + 4.09i)15-s − 5·16-s − 7.73·17-s + (−3.86 + 6.69i)18-s + (−1 + 1.73i)19-s + ⋯
L(s)  = 1  + 1.22·2-s + (−0.788 − 1.36i)3-s + 0.499·4-s + (−0.387 − 0.670i)5-s + (−0.965 − 1.67i)6-s − 0.612·8-s + (−0.744 + 1.28i)9-s + (−0.474 − 0.821i)10-s + (−0.191 − 0.331i)11-s + (−0.394 − 0.683i)12-s + (0.997 − 0.0643i)13-s + (−0.610 + 1.05i)15-s − 1.25·16-s − 1.87·17-s + (−0.911 + 1.57i)18-s + (−0.229 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.997 - 0.0769i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.997 - 0.0769i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0429845 + 1.11510i\)
\(L(\frac12)\) \(\approx\) \(0.0429845 + 1.11510i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.59 + 0.232i)T \)
good2 \( 1 - 1.73T + 2T^{2} \)
3 \( 1 + (1.36 + 2.36i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.866 + 1.5i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.633 + 1.09i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 7.73T + 17T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.73T + 23T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.09 - 3.63i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (-2.59 + 4.5i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.0980 - 0.169i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.46 + 11.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.96 + 8.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.26T + 59T^{2} \)
61 \( 1 + (-2.40 + 4.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.09 - 5.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.59 + 2.76i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.09 + 14.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.19T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 + (3.19 + 5.53i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.65839249535355998155357053965, −8.825368961254140536317773013236, −8.422976872643025250980281385259, −6.97786485916507294321313482141, −6.45743192805692599652003833837, −5.53203478773106523627488391067, −4.74434086558219761284269671910, −3.61821477540191044518639051487, −2.06608907513741665208850473742, −0.43518496439568744079557733255, 2.82887423295651320510993894621, 3.85535833275129304634712158863, 4.49621613864440563568465439889, 5.26901309716399218368244729722, 6.26678340691436440738378437100, 6.98081776627880857927565488118, 8.719174848114114697564545179212, 9.358930521813043969891911470372, 10.59489996485731797900938517087, 11.15934091577868534003888812564

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