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Properties

Label 2-637-91.45-c1-0-21
Degree $2$
Conductor $637$
Sign $0.970 - 0.242i$
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  + (−0.976 + 0.976i)2-s + (−0.928 + 0.536i)3-s + 0.0940i·4-s + (0.742 − 2.77i)5-s + (0.383 − 1.42i)6-s + (−2.04 − 2.04i)8-s + (−0.925 + 1.60i)9-s + (1.98 + 3.43i)10-s + (−0.369 + 1.37i)11-s + (−0.0504 − 0.0873i)12-s + (−3.54 − 0.634i)13-s + (0.796 + 2.97i)15-s + 3.80·16-s + 4.19·17-s + (−0.661 − 2.46i)18-s + (5.95 − 1.59i)19-s + ⋯
L(s)  = 1  + (−0.690 + 0.690i)2-s + (−0.536 + 0.309i)3-s + 0.0470i·4-s + (0.332 − 1.23i)5-s + (0.156 − 0.583i)6-s + (−0.722 − 0.722i)8-s + (−0.308 + 0.534i)9-s + (0.626 + 1.08i)10-s + (−0.111 + 0.415i)11-s + (−0.0145 − 0.0252i)12-s + (−0.984 − 0.176i)13-s + (0.205 + 0.767i)15-s + 0.950·16-s + 1.01·17-s + (−0.155 − 0.581i)18-s + (1.36 − 0.366i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762048 + 0.0936953i\)
\(L(\frac12)\) \(\approx\) \(0.762048 + 0.0936953i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.54 + 0.634i)T \)
good2 \( 1 + (0.976 - 0.976i)T - 2iT^{2} \)
3 \( 1 + (0.928 - 0.536i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.742 + 2.77i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.369 - 1.37i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 4.19T + 17T^{2} \)
19 \( 1 + (-5.95 + 1.59i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 7.82iT - 23T^{2} \)
29 \( 1 + (-0.441 + 0.764i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.886 - 0.237i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-5.26 - 5.26i)T + 37iT^{2} \)
41 \( 1 + (-11.4 + 3.07i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.809 + 0.467i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.01 - 0.808i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.26 - 2.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.13 + 4.13i)T - 59iT^{2} \)
61 \( 1 + (0.0739 + 0.0427i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.995 + 0.266i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.79 - 0.750i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.737 + 2.75i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.71 + 8.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.54 + 1.54i)T + 83iT^{2} \)
89 \( 1 + (3.48 - 3.48i)T - 89iT^{2} \)
97 \( 1 + (-2.37 + 8.87i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.22964035395776020837843747008, −9.637521421327291505114726257350, −8.855872302037531683708255366402, −7.960013617180472161854856085578, −7.35045018337805881515774893189, −6.04269923820076863086210473980, −5.21678224515084601079389911284, −4.46309258386532417215216331315, −2.72660272921823896395754735970, −0.70499680770666218646120602920, 1.08231216032284397732413595636, 2.58435050874511909455275228983, 3.41490086954779292135962106097, 5.50876396283645586967771579922, 5.87996529019981506973131899553, 7.08925476700170397053191137966, 7.81648062953359737238978978265, 9.342775359555601435834566134885, 9.688910168512532963798307546604, 10.56864036288541358279754684300

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