Properties

Label 2-637-49.37-c1-0-39
Degree $2$
Conductor $637$
Sign $0.727 + 0.686i$
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  + (−0.971 + 0.662i)2-s + (1.44 − 1.34i)3-s + (−0.225 + 0.575i)4-s + (−0.154 − 0.0475i)5-s + (−0.515 + 2.25i)6-s + (2.49 − 0.876i)7-s + (−0.684 − 3.00i)8-s + (0.0659 − 0.880i)9-s + (0.181 − 0.0559i)10-s + (−0.385 − 5.14i)11-s + (0.444 + 1.13i)12-s + (0.900 + 0.433i)13-s + (−1.84 + 2.50i)14-s + (−0.286 + 0.138i)15-s + (1.74 + 1.62i)16-s + (−4.86 − 0.733i)17-s + ⋯
L(s)  = 1  + (−0.686 + 0.468i)2-s + (0.833 − 0.773i)3-s + (−0.112 + 0.287i)4-s + (−0.0689 − 0.0212i)5-s + (−0.210 + 0.922i)6-s + (0.943 − 0.331i)7-s + (−0.242 − 1.06i)8-s + (0.0219 − 0.293i)9-s + (0.0573 − 0.0176i)10-s + (−0.116 − 1.55i)11-s + (0.128 + 0.327i)12-s + (0.249 + 0.120i)13-s + (−0.493 + 0.669i)14-s + (−0.0740 + 0.0356i)15-s + (0.436 + 0.405i)16-s + (−1.17 − 0.177i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.727 + 0.686i$
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: \((2,\ 637,\ (\ :1/2),\ 0.727 + 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21942 - 0.484815i\)
\(L(\frac12)\) \(\approx\) \(1.21942 - 0.484815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.49 + 0.876i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
good2 \( 1 + (0.971 - 0.662i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (-1.44 + 1.34i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (0.154 + 0.0475i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (0.385 + 5.14i)T + (-10.8 + 1.63i)T^{2} \)
17 \( 1 + (4.86 + 0.733i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-2.26 + 3.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.27 + 0.192i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (-2.10 + 2.64i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-4.23 - 7.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.63 - 6.71i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (1.41 + 6.21i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-2.77 + 12.1i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (2.44 - 1.66i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-0.410 + 1.04i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-0.454 + 0.140i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (1.73 + 4.41i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (0.0180 + 0.0312i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.16 - 6.48i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (1.48 + 1.01i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (5.95 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-13.4 + 6.46i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.583 - 7.78i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 5.41T + 97T^{2} \)
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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

−10.41704443549041878007339928754, −9.035073849361616767482199087423, −8.532649894567673545390406673303, −8.068434453339555714891150733337, −7.19604235117467347814201807426, −6.44770949129360417198407923377, −4.93775132259173769719707776679, −3.66070171193306449639680506385, −2.50716328302394733894963964627, −0.879604635678906885791495298747, 1.65151268779331419780648272990, 2.62256909913231825523225906504, 4.18871061621359308692546167799, 4.86755376083103416735961634267, 6.09848664460028104712044521828, 7.66090402408794415077527638122, 8.281766460626222163808323015026, 9.262511409411653798267344305613, 9.626406696962729176960826411401, 10.45303783196751990678936680556

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