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Properties

Label 2-637-91.23-c1-0-7
Degree $2$
Conductor $637$
Sign $-0.981 + 0.189i$
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  + 2.58i·2-s + (0.259 − 0.449i)3-s − 4.70·4-s + (1.39 + 0.806i)5-s + (1.16 + 0.671i)6-s − 6.99i·8-s + (1.36 + 2.36i)9-s + (−2.08 + 3.61i)10-s + (2.34 + 1.35i)11-s + (−1.21 + 2.11i)12-s + (−2.36 + 2.71i)13-s + (0.723 − 0.417i)15-s + 8.69·16-s + 3.12·17-s + (−6.12 + 3.53i)18-s + (−3.18 + 1.84i)19-s + ⋯
L(s)  = 1  + 1.83i·2-s + (0.149 − 0.259i)3-s − 2.35·4-s + (0.624 + 0.360i)5-s + (0.474 + 0.273i)6-s − 2.47i·8-s + (0.455 + 0.788i)9-s + (−0.659 + 1.14i)10-s + (0.706 + 0.407i)11-s + (−0.351 + 0.609i)12-s + (−0.656 + 0.753i)13-s + (0.186 − 0.107i)15-s + 2.17·16-s + 0.758·17-s + (−1.44 + 0.833i)18-s + (−0.731 + 0.422i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.981 + 0.189i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.981 + 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.133519 - 1.39418i\)
\(L(\frac12)\) \(\approx\) \(0.133519 - 1.39418i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.36 - 2.71i)T \)
good2 \( 1 - 2.58iT - 2T^{2} \)
3 \( 1 + (-0.259 + 0.449i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.39 - 0.806i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.34 - 1.35i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.12T + 17T^{2} \)
19 \( 1 + (3.18 - 1.84i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.98T + 23T^{2} \)
29 \( 1 + (-2.68 - 4.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (9.07 - 5.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.95iT - 37T^{2} \)
41 \( 1 + (-6.66 + 3.85i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.67 - 2.90i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.913 + 0.527i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.63 + 6.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 11.4iT - 59T^{2} \)
61 \( 1 + (1.46 + 2.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.7 - 6.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.17 - 0.675i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.88 + 4.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.10 + 5.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.69iT - 83T^{2} \)
89 \( 1 - 1.75iT - 89T^{2} \)
97 \( 1 + (-13.4 - 7.74i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.69483182345599992039326196523, −9.805621896318692973851634948830, −9.098793352743698306682796923932, −8.151477962192414011172463059145, −7.27080813444731917476507422068, −6.78198900679245237612170341609, −5.84219715830201404193525699508, −4.93476340073072764356582877754, −3.96050445208946493191239997078, −1.94211240745997086763732667232, 0.793493679917535543387455334819, 2.05303987804908237864856313611, 3.27469693279845228844432075586, 4.08923989571851463249925153024, 5.13077070349787715490189604508, 6.24668674200776192403950545026, 7.87324272189694621114847030001, 8.963925331897294553324677857975, 9.598481166752857452531609813536, 10.00185559597377109650137157290

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