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Properties

Label 2-637-91.16-c1-0-30
Degree $2$
Conductor $637$
Sign $0.721 - 0.692i$
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  + 2.61·2-s + (−1.30 + 2.26i)3-s + 4.85·4-s + (1.30 − 2.26i)5-s + (−3.42 + 5.93i)6-s + 7.47·8-s + (−1.92 − 3.33i)9-s + (3.42 − 5.93i)10-s + (−0.927 + 1.60i)11-s + (−6.35 + 11.0i)12-s + (2.5 + 2.59i)13-s + (3.42 + 5.93i)15-s + 9.85·16-s + 1.47·17-s + (−5.04 − 8.73i)18-s + (−0.927 − 1.60i)19-s + ⋯
L(s)  = 1  + 1.85·2-s + (−0.755 + 1.30i)3-s + 2.42·4-s + (0.585 − 1.01i)5-s + (−1.39 + 2.42i)6-s + 2.64·8-s + (−0.642 − 1.11i)9-s + (1.08 − 1.87i)10-s + (−0.279 + 0.484i)11-s + (−1.83 + 3.17i)12-s + (0.693 + 0.720i)13-s + (0.884 + 1.53i)15-s + 2.46·16-s + 0.357·17-s + (−1.18 − 2.05i)18-s + (−0.212 − 0.368i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.721 - 0.692i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.721 - 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.54566 + 1.42498i\)
\(L(\frac12)\) \(\approx\) \(3.54566 + 1.42498i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 2.59i)T \)
good2 \( 1 - 2.61T + 2T^{2} \)
3 \( 1 + (1.30 - 2.26i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.30 + 2.26i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.927 - 1.60i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 1.47T + 17T^{2} \)
19 \( 1 + (0.927 + 1.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.47T + 23T^{2} \)
29 \( 1 + (3.54 + 6.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.35 + 4.07i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (-0.381 - 0.661i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.28 - 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.11 - 1.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.88 + 3.25i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.23T + 59T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.35 + 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.09 + 12.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + 4.90T + 89T^{2} \)
97 \( 1 + (-9.42 + 16.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.12872504092354627083960733322, −9.964032974578359402097012515410, −9.362488611552687845597493504183, −7.87481137489538828707818936593, −6.35814881585428269160385938384, −5.82341529911665745838019732906, −4.93219282137661721455739809394, −4.45663315585409413447069379874, −3.60465091763564915509551465884, −1.96810569160220177028483107403, 1.66340222706922530234289163376, 2.76883349772188162769633074744, 3.75982478259556372361563691050, 5.54502247221733473876116968591, 5.69975358444092434768934146623, 6.66703667723165291746282529119, 7.14433896343688245098706975224, 8.217967955127487808178811589000, 10.28647118629895280768246124579, 10.87607279899854349351159835297

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