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Properties

Label 2-637-49.37-c1-0-36
Degree $2$
Conductor $637$
Sign $0.929 + 0.367i$
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  + (1.01 − 0.694i)2-s + (0.670 − 0.621i)3-s + (−0.175 + 0.447i)4-s + (0.228 + 0.0704i)5-s + (0.250 − 1.09i)6-s + (2.28 − 1.33i)7-s + (0.680 + 2.98i)8-s + (−0.161 + 2.15i)9-s + (0.281 − 0.0867i)10-s + (−0.0880 − 1.17i)11-s + (0.160 + 0.409i)12-s + (0.900 + 0.433i)13-s + (1.40 − 2.94i)14-s + (0.196 − 0.0947i)15-s + (2.05 + 1.90i)16-s + (4.17 + 0.629i)17-s + ⋯
L(s)  = 1  + (0.720 − 0.490i)2-s + (0.386 − 0.359i)3-s + (−0.0878 + 0.223i)4-s + (0.102 + 0.0314i)5-s + (0.102 − 0.448i)6-s + (0.863 − 0.503i)7-s + (0.240 + 1.05i)8-s + (−0.0539 + 0.719i)9-s + (0.0889 − 0.0274i)10-s + (−0.0265 − 0.354i)11-s + (0.0463 + 0.118i)12-s + (0.249 + 0.120i)13-s + (0.374 − 0.786i)14-s + (0.0508 − 0.0244i)15-s + (0.514 + 0.477i)16-s + (1.01 + 0.152i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.929 + 0.367i$
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: \((2,\ 637,\ (\ :1/2),\ 0.929 + 0.367i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.57785 - 0.491180i\)
\(L(\frac12)\) \(\approx\) \(2.57785 - 0.491180i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.28 + 1.33i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
good2 \( 1 + (-1.01 + 0.694i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (-0.670 + 0.621i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (-0.228 - 0.0704i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (0.0880 + 1.17i)T + (-10.8 + 1.63i)T^{2} \)
17 \( 1 + (-4.17 - 0.629i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-1.03 + 1.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.137 + 0.0207i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (-1.31 + 1.65i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (0.459 + 0.796i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.24 + 3.18i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (1.38 + 6.08i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (2.52 - 11.0i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (5.15 - 3.51i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-3.19 + 8.14i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (1.16 - 0.360i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-1.30 - 3.32i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (4.41 + 7.64i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.30 + 5.39i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-10.1 - 6.92i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (6.45 - 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (12.4 - 6.01i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-0.412 + 5.49i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−10.87175712956248375077120147931, −9.803847273426993204450270110851, −8.404268168252444582488740860144, −8.063589289252945531406255971464, −7.18566665429825766418714997138, −5.69262292642180816721938301333, −4.83283764755869535947301030332, −3.87102023187963879277892105065, −2.76326510518006182506170228329, −1.63532147465774795800073496173, 1.44311628946797055342165788218, 3.22269491360388983342191202993, 4.19714555481672678580912440279, 5.22017407357919849059934317992, 5.85281893468493662841197127178, 6.94278797123665073486699815530, 7.972050797726536786738416116316, 8.933764194250462537405200187028, 9.757571632909510151364073096989, 10.42106174755649126931856954710

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