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Properties

Label 2-637-91.51-c1-0-16
Degree $2$
Conductor $637$
Sign $-0.998 - 0.0623i$
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 − i)2-s + (−1 − 1.73i)3-s + (0.999 + 1.73i)4-s + (0.866 + 0.5i)5-s + 3.99i·6-s + (−0.499 + 0.866i)9-s + (−0.999 − 1.73i)10-s + (1.73 − i)11-s + (2 − 3.46i)12-s + (2 + 3i)13-s − 1.99i·15-s + (1.99 − 3.46i)16-s + (−3 − 5.19i)17-s + (1.73 − 0.999i)18-s + (−2.59 − 1.5i)19-s + 1.99i·20-s + ⋯
L(s)  = 1  + (−1.22 − 0.707i)2-s + (−0.577 − 0.999i)3-s + (0.499 + 0.866i)4-s + (0.387 + 0.223i)5-s + 1.63i·6-s + (−0.166 + 0.288i)9-s + (−0.316 − 0.547i)10-s + (0.522 − 0.301i)11-s + (0.577 − 0.999i)12-s + (0.554 + 0.832i)13-s − 0.516i·15-s + (0.499 − 0.866i)16-s + (−0.727 − 1.26i)17-s + (0.408 − 0.235i)18-s + (−0.596 − 0.344i)19-s + 0.447i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.998 - 0.0623i$
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: \((2,\ 637,\ (\ :1/2),\ -0.998 - 0.0623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0162020 + 0.519572i\)
\(L(\frac12)\) \(\approx\) \(0.0162020 + 0.519572i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 3i)T \)
good2 \( 1 + (1.73 + i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.73 + i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.59 + 1.5i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (-2.59 + 1.5i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.19 - 3i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (9.52 + 5.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.92 - 4i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 - 6i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14iT - 71T^{2} \)
73 \( 1 + (7.79 - 4.5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11iT - 83T^{2} \)
89 \( 1 + (-4.33 - 2.5i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9iT - 97T^{2} \)
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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

−10.20266025074205422863379871231, −9.176827011094689127318373783099, −8.712073305864763765335015901083, −7.51759282157878990157575312575, −6.68845619352069290050098515288, −6.00957795600349529464911213867, −4.48574036799761620071783583466, −2.70431623710146815301769150790, −1.68544795204368746326145464150, −0.48845699821617452240882744783, 1.49192309066383868113974824918, 3.66694748601762253733818314153, 4.69640184917795575870205671030, 5.91347135671435733127593712457, 6.45528690673301836786628203947, 7.77348869076820292387597583805, 8.477066108464837108155107626347, 9.400678910813440418796796718524, 9.949136386856369404129814704666, 10.69523471130135977651715371211

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