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Properties

Label 2-637-91.30-c1-0-31
Degree $2$
Conductor $637$
Sign $0.990 + 0.139i$
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  + (1.20 − 0.692i)2-s + 2.82·3-s + (−0.0395 + 0.0685i)4-s + (0.449 + 0.259i)5-s + (3.39 − 1.95i)6-s + 2.88i·8-s + 4.98·9-s + 0.719·10-s − 1.62i·11-s + (−0.111 + 0.193i)12-s + (−1.42 + 3.31i)13-s + (1.26 + 0.733i)15-s + (1.91 + 3.32i)16-s + (−0.974 + 1.68i)17-s + (5.98 − 3.45i)18-s − 2.49i·19-s + ⋯
L(s)  = 1  + (0.848 − 0.490i)2-s + 1.63·3-s + (−0.0197 + 0.0342i)4-s + (0.200 + 0.116i)5-s + (1.38 − 0.799i)6-s + 1.01i·8-s + 1.66·9-s + 0.227·10-s − 0.489i·11-s + (−0.0322 + 0.0559i)12-s + (−0.395 + 0.918i)13-s + (0.327 + 0.189i)15-s + (0.479 + 0.830i)16-s + (−0.236 + 0.409i)17-s + (1.41 − 0.814i)18-s − 0.571i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.990 + 0.139i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.990 + 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.62448 - 0.254883i\)
\(L(\frac12)\) \(\approx\) \(3.62448 - 0.254883i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 - 3.31i)T \)
good2 \( 1 + (-1.20 + 0.692i)T + (1 - 1.73i)T^{2} \)
3 \( 1 - 2.82T + 3T^{2} \)
5 \( 1 + (-0.449 - 0.259i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 1.62iT - 11T^{2} \)
17 \( 1 + (0.974 - 1.68i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 2.49iT - 19T^{2} \)
23 \( 1 + (4.57 + 7.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.61 + 4.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.01 - 2.89i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.85 + 5.11i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.64 + 2.10i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.91 - 2.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.44 - 7.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.37 + 3.10i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 - 8.37iT - 67T^{2} \)
71 \( 1 + (4.50 - 2.59i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (10.2 - 5.91i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.491 - 0.850i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.91iT - 83T^{2} \)
89 \( 1 + (10.4 - 6.00i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.82 - 2.21i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.55549047767982973865917057461, −9.584112649876406346729261297476, −8.659832461896696608838775823189, −8.248177851195752394341917514012, −7.14820594187040887454499629485, −5.94250744005653686451773468311, −4.39228571378276021824142774553, −3.97592788311964284831255894918, −2.66655363001643539928970403973, −2.20275960546795291364204933941, 1.73773366430349332406838130074, 3.13434006623487166524675821770, 3.91162640611245719808539159057, 5.02139219320556558079036361079, 5.94959784929095635991150789398, 7.29303859915558139765667561919, 7.74167294064053313560415900339, 8.872250375804771905492233256606, 9.759758253252924660926671128103, 10.06402848968646398000937721545

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