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Properties

Label 2-637-91.31-c1-0-23
Degree $2$
Conductor $637$
Sign $0.106 - 0.994i$
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  + (1.98 + 0.531i)2-s + (1.57 + 0.909i)3-s + (1.91 + 1.10i)4-s + (−0.737 + 2.75i)5-s + (2.64 + 2.64i)6-s + (0.311 + 0.311i)8-s + (0.155 + 0.269i)9-s + (−2.92 + 5.06i)10-s + (−0.616 + 0.165i)11-s + (2.01 + 3.48i)12-s + (3.40 + 1.19i)13-s + (−3.66 + 3.66i)15-s + (−1.76 − 3.05i)16-s + (2.16 − 3.74i)17-s + (0.165 + 0.616i)18-s + (−1.24 + 4.64i)19-s + ⋯
L(s)  = 1  + (1.40 + 0.375i)2-s + (0.909 + 0.525i)3-s + (0.958 + 0.553i)4-s + (−0.329 + 1.23i)5-s + (1.07 + 1.07i)6-s + (0.109 + 0.109i)8-s + (0.0518 + 0.0898i)9-s + (−0.924 + 1.60i)10-s + (−0.186 + 0.0498i)11-s + (0.581 + 1.00i)12-s + (0.943 + 0.330i)13-s + (−0.946 + 0.946i)15-s + (−0.440 − 0.763i)16-s + (0.524 − 0.908i)17-s + (0.0389 + 0.145i)18-s + (−0.285 + 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.106 - 0.994i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.106 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.78526 + 2.50319i\)
\(L(\frac12)\) \(\approx\) \(2.78526 + 2.50319i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.40 - 1.19i)T \)
good2 \( 1 + (-1.98 - 0.531i)T + (1.73 + i)T^{2} \)
3 \( 1 + (-1.57 - 0.909i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.737 - 2.75i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.616 - 0.165i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.16 + 3.74i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.24 - 4.64i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.808 + 0.466i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.33T + 29T^{2} \)
31 \( 1 + (7.48 - 2.00i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.783 + 2.92i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.81 - 1.81i)T + 41iT^{2} \)
43 \( 1 + 10.4iT - 43T^{2} \)
47 \( 1 + (8.07 + 2.16i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.68 - 2.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.0935 + 0.349i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.74 + 3.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.66 - 9.94i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-5.56 + 5.56i)T - 71iT^{2} \)
73 \( 1 + (3.24 + 12.1i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.89 - 11.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.30 - 4.30i)T + 83iT^{2} \)
89 \( 1 + (7.66 + 2.05i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.236 + 0.236i)T + 97iT^{2} \)
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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

−10.88537313175216411594988528130, −9.978177547914400315018461749941, −9.024265998128497484635813915017, −7.972968949884646476122681970989, −6.98447447260026637492334660493, −6.26894821877490601895686888400, −5.19938868049235101841491947008, −3.88054614515744371744322112704, −3.48718879635302663482223316251, −2.56955860812261532399028333998, 1.44604788865539915507623225878, 2.76308784051988641525330237527, 3.71292346060938364653093651539, 4.69632602428741912070087026928, 5.50611304870835348184143102557, 6.58344877366112273325772604889, 8.017585594679051511374956380931, 8.435362008746821782828862982629, 9.267665037597403719319917332785, 10.75337241106349470905431240421

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