Properties

Label 2-637-49.37-c1-0-17
Degree $2$
Conductor $637$
Sign $0.0488 - 0.998i$
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  + (−2.30 + 1.57i)2-s + (0.887 − 0.823i)3-s + (2.12 − 5.41i)4-s + (−0.535 − 0.165i)5-s + (−0.753 + 3.29i)6-s + (2.63 + 0.247i)7-s + (2.37 + 10.4i)8-s + (−0.114 + 1.53i)9-s + (1.49 − 0.462i)10-s + (0.286 + 3.82i)11-s + (−2.57 − 6.55i)12-s + (−0.900 − 0.433i)13-s + (−6.47 + 3.57i)14-s + (−0.611 + 0.294i)15-s + (−13.3 − 12.3i)16-s + (4.09 + 0.616i)17-s + ⋯
L(s)  = 1  + (−1.63 + 1.11i)2-s + (0.512 − 0.475i)3-s + (1.06 − 2.70i)4-s + (−0.239 − 0.0739i)5-s + (−0.307 + 1.34i)6-s + (0.995 + 0.0937i)7-s + (0.839 + 3.67i)8-s + (−0.0382 + 0.510i)9-s + (0.473 − 0.146i)10-s + (0.0863 + 1.15i)11-s + (−0.742 − 1.89i)12-s + (−0.249 − 0.120i)13-s + (−1.73 + 0.955i)14-s + (−0.157 + 0.0760i)15-s + (−3.33 − 3.09i)16-s + (0.992 + 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0488 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0488 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.0488 - 0.998i$
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: \((2,\ 637,\ (\ :1/2),\ 0.0488 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.597819 + 0.569322i\)
\(L(\frac12)\) \(\approx\) \(0.597819 + 0.569322i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.63 - 0.247i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
good2 \( 1 + (2.30 - 1.57i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (-0.887 + 0.823i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (0.535 + 0.165i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (-0.286 - 3.82i)T + (-10.8 + 1.63i)T^{2} \)
17 \( 1 + (-4.09 - 0.616i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-1.37 + 2.38i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.27 - 0.644i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (2.87 - 3.60i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-4.15 - 7.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.52 - 6.42i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (1.10 + 4.81i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (0.0580 - 0.254i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (2.06 - 1.41i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-3.80 + 9.70i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-6.82 + 2.10i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-5.35 - 13.6i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (0.00818 + 0.0141i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.51 - 8.16i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (4.34 + 2.96i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (-2.98 + 5.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.99 - 3.84i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.439 - 5.86i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−10.28902407633603415582618920994, −9.839135200348310853761107789648, −8.627019598060024497705820262831, −8.180947271646241452591309265855, −7.45966224339433393943176288576, −6.93788591225205750975271727666, −5.55820042193896619316538528130, −4.79503431129677679819422757646, −2.28383878033940906863460254349, −1.38070235525058716644672965731, 0.807122059342974109901912011761, 2.23374499155953242738585930213, 3.43951219943624557672967403695, 4.06883157519349261766666837890, 6.05896728785664634604671130833, 7.61422503211305170096809132432, 7.987054600769779184886314489742, 8.766932903538956657432502777026, 9.639027262107433997045556410590, 10.11504564888205152128859435007

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