Properties

Label 2-637-91.4-c1-0-36
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: $\chi_{637} (459, \cdot )$
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

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

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