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Properties

Label 2-637-91.51-c1-0-5
Degree $2$
Conductor $637$
Sign $-0.0288 - 0.999i$
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  + (−0.589 − 0.340i)2-s + (1.16 + 2.02i)3-s + (−0.768 − 1.33i)4-s + (2.81 + 1.62i)5-s − 1.58i·6-s + 2.40i·8-s + (−1.22 + 2.12i)9-s + (−1.10 − 1.91i)10-s + (−4.61 + 2.66i)11-s + (1.79 − 3.10i)12-s + (−1.47 + 3.28i)13-s + 7.57i·15-s + (−0.719 + 1.24i)16-s + (1.33 + 2.31i)17-s + (1.44 − 0.832i)18-s + (0.603 + 0.348i)19-s + ⋯
L(s)  = 1  + (−0.416 − 0.240i)2-s + (0.673 + 1.16i)3-s + (−0.384 − 0.665i)4-s + (1.25 + 0.725i)5-s − 0.648i·6-s + 0.850i·8-s + (−0.408 + 0.706i)9-s + (−0.349 − 0.604i)10-s + (−1.39 + 0.803i)11-s + (0.517 − 0.897i)12-s + (−0.409 + 0.912i)13-s + 1.95i·15-s + (−0.179 + 0.311i)16-s + (0.323 + 0.560i)17-s + (0.339 − 0.196i)18-s + (0.138 + 0.0798i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0288 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0288 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.0288 - 0.999i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.0288 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00932 + 1.03883i\)
\(L(\frac12)\) \(\approx\) \(1.00932 + 1.03883i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.47 - 3.28i)T \)
good2 \( 1 + (0.589 + 0.340i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.16 - 2.02i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.81 - 1.62i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.61 - 2.66i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.33 - 2.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.603 - 0.348i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.48 + 7.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.28T + 29T^{2} \)
31 \( 1 + (4.72 - 2.72i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.79 - 3.34i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.21iT - 41T^{2} \)
43 \( 1 - 2.39T + 43T^{2} \)
47 \( 1 + (-5.01 - 2.89i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.35 - 4.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.58 + 1.49i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.09 - 1.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.53 - 4.35i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.56iT - 71T^{2} \)
73 \( 1 + (-5.09 + 2.94i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.455 - 0.788i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.7iT - 83T^{2} \)
89 \( 1 + (0.853 + 0.493i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.3iT - 97T^{2} \)
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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

−10.48160345312004745867096839701, −9.955172633641984920696982660676, −9.339028401889304912942545048283, −8.688178478957535818280287039106, −7.37194788078463146254405263197, −6.14732582936973673740137205909, −5.16437107601114317322603417201, −4.40220320042984602099656700696, −2.80449264280363212037459411960, −2.01462032186154419697467580040, 0.843066296923914158251080585200, 2.34102100385921741397719189327, 3.27429389076097371863563185268, 5.16796955066610446397716172034, 5.77895067516158486916554588111, 7.33842587988091217077907089366, 7.64978912993693921844420233349, 8.538109586112021022212770970619, 9.275868511615773203475103409500, 9.959162508057231844869883818528

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