Properties

Label 2-637-49.4-c1-0-41
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  + (1.73 + 1.17i)2-s + (−0.623 − 0.578i)3-s + (0.871 + 2.22i)4-s + (3.60 − 1.11i)5-s + (−0.396 − 1.73i)6-s + (−2.15 − 1.53i)7-s + (−0.179 + 0.788i)8-s + (−0.170 − 2.27i)9-s + (7.54 + 2.32i)10-s + (−0.324 + 4.32i)11-s + (0.741 − 1.88i)12-s + (0.900 − 0.433i)13-s + (−1.91 − 5.20i)14-s + (−2.88 − 1.39i)15-s + (2.25 − 2.09i)16-s + (4.97 − 0.749i)17-s + ⋯
L(s)  = 1  + (1.22 + 0.834i)2-s + (−0.359 − 0.333i)3-s + (0.435 + 1.11i)4-s + (1.61 − 0.497i)5-s + (−0.161 − 0.708i)6-s + (−0.813 − 0.581i)7-s + (−0.0636 + 0.278i)8-s + (−0.0567 − 0.756i)9-s + (2.38 + 0.736i)10-s + (−0.0978 + 1.30i)11-s + (0.214 − 0.545i)12-s + (0.249 − 0.120i)13-s + (−0.510 − 1.39i)14-s + (−0.746 − 0.359i)15-s + (0.564 − 0.523i)16-s + (1.20 − 0.181i)17-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: $\chi_{637} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.981 - 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.00892 + 0.288389i\)
\(L(\frac12)\) \(\approx\) \(3.00892 + 0.288389i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.15 + 1.53i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
good2 \( 1 + (-1.73 - 1.17i)T + (0.730 + 1.86i)T^{2} \)
3 \( 1 + (0.623 + 0.578i)T + (0.224 + 2.99i)T^{2} \)
5 \( 1 + (-3.60 + 1.11i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (0.324 - 4.32i)T + (-10.8 - 1.63i)T^{2} \)
17 \( 1 + (-4.97 + 0.749i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (0.0277 + 0.0481i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.21 + 0.785i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (-0.928 - 1.16i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (4.14 - 7.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.30 - 5.86i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (0.0523 - 0.229i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (1.73 + 7.59i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-5.90 - 4.02i)T + (17.1 + 43.7i)T^{2} \)
53 \( 1 + (0.819 + 2.08i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (2.02 + 0.623i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (5.27 - 13.4i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (-1.79 + 3.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.70 - 7.14i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (10.9 - 7.45i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (5.44 + 9.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-13.0 - 6.30i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (1.04 + 13.9i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−10.17055792686497331277058853844, −9.992791255331987221993145683913, −8.945102912895401538693056570339, −7.36094236037682055457825468469, −6.76392788869769054719984013070, −5.94766847835429281631671821158, −5.43153296261126438167185287595, −4.34098975047605459422722145848, −3.14377967533490496702187783334, −1.40245225647117043479664339283, 1.93047482042736034711387146435, 2.82028642863012973403231892529, 3.74839600266339167699149900897, 5.24559277359207643914617214750, 5.88522036697829457070973763580, 6.12961541365382981437602725727, 7.943717031307235796756812305904, 9.235946732083203143214286195393, 10.06978341774645067004375855215, 10.64904556605916466134924492606

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