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Properties

Label 2-637-91.74-c1-0-3
Degree $2$
Conductor $637$
Sign $0.855 + 0.517i$
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  − 1.30·2-s + (−1.44 − 2.49i)3-s − 0.302·4-s + (−1.44 − 2.49i)5-s + (1.87 + 3.25i)6-s + 3·8-s + (−2.65 + 4.59i)9-s + (1.87 + 3.25i)10-s + (2.95 + 5.11i)11-s + (0.436 + 0.755i)12-s + (3.31 + 1.41i)13-s + (−4.15 + 7.19i)15-s − 3.30·16-s − 0.872·17-s + (3.45 − 5.98i)18-s + (−1.44 + 2.49i)19-s + ⋯
L(s)  = 1  − 0.921·2-s + (−0.831 − 1.44i)3-s − 0.151·4-s + (−0.644 − 1.11i)5-s + (0.766 + 1.32i)6-s + 1.06·8-s + (−0.883 + 1.53i)9-s + (0.593 + 1.02i)10-s + (0.890 + 1.54i)11-s + (0.125 + 0.218i)12-s + (0.920 + 0.391i)13-s + (−1.07 + 1.85i)15-s − 0.825·16-s − 0.211·17-s + (0.814 − 1.41i)18-s + (−0.330 + 0.572i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.855 + 0.517i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.855 + 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.466384 - 0.130024i\)
\(L(\frac12)\) \(\approx\) \(0.466384 - 0.130024i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 - 1.41i)T \)
good2 \( 1 + 1.30T + 2T^{2} \)
3 \( 1 + (1.44 + 2.49i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.44 + 2.49i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.95 - 5.11i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.872T + 17T^{2} \)
19 \( 1 + (1.44 - 2.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.60T + 23T^{2} \)
29 \( 1 + (0.651 - 1.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.436 - 0.755i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.39T + 37T^{2} \)
41 \( 1 + (3.75 - 6.50i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.75 + 4.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.19 - 10.7i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.80 - 8.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.63T + 59T^{2} \)
61 \( 1 + (2.88 - 4.99i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2 + 3.46i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.88 + 4.99i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.302 + 0.524i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.63T + 83T^{2} \)
89 \( 1 - 8.64T + 89T^{2} \)
97 \( 1 + (-3.88 - 6.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.58098492257447477595484163011, −9.310774915865019746523057445773, −8.746605358177279788840623326297, −7.83397448089972946884450358315, −7.17892964567437183333514514018, −6.32812567060376394619412090400, −4.97190726669905956696375664226, −4.21614897193416035466260421634, −1.67215533806631131637059348646, −1.02926245911253534949341727189, 0.56274131094727279762342466635, 3.34516918117579994451855229286, 3.91763508375005680301039835934, 5.11242223815647225869610372543, 6.19374465385840783759406808372, 7.09669228964807819651706282036, 8.500928515868574203546895596723, 8.913519765712631460114741829223, 9.935898326429240579224384726168, 10.71523846634753745410436444068

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