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Properties

Label 2-637-91.74-c1-0-22
Degree $2$
Conductor $637$
Sign $0.927 + 0.374i$
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  + 0.414·2-s + (0.707 + 1.22i)3-s − 1.82·4-s + (−0.914 − 1.58i)5-s + (0.292 + 0.507i)6-s − 1.58·8-s + (0.500 − 0.866i)9-s + (−0.378 − 0.655i)10-s + (−0.292 − 0.507i)11-s + (−1.29 − 2.23i)12-s + (3.5 + 0.866i)13-s + (1.29 − 2.23i)15-s + 3·16-s + 5.82·17-s + (0.207 − 0.358i)18-s + (3 − 5.19i)19-s + ⋯
L(s)  = 1  + 0.292·2-s + (0.408 + 0.707i)3-s − 0.914·4-s + (−0.408 − 0.708i)5-s + (0.119 + 0.207i)6-s − 0.560·8-s + (0.166 − 0.288i)9-s + (−0.119 − 0.207i)10-s + (−0.0883 − 0.152i)11-s + (−0.373 − 0.646i)12-s + (0.970 + 0.240i)13-s + (0.333 − 0.578i)15-s + 0.750·16-s + 1.41·17-s + (0.0488 − 0.0845i)18-s + (0.688 − 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.927 + 0.374i$
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: \((2,\ 637,\ (\ :1/2),\ 0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50497 - 0.292578i\)
\(L(\frac12)\) \(\approx\) \(1.50497 - 0.292578i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 0.866i)T \)
good2 \( 1 - 0.414T + 2T^{2} \)
3 \( 1 + (-0.707 - 1.22i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.914 + 1.58i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.292 + 0.507i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.82T + 17T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 + (2.08 - 3.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.29 + 2.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.48T + 37T^{2} \)
41 \( 1 + (-0.0857 + 0.148i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.70 + 2.95i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.82 + 3.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + (-2.08 + 3.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.12 + 3.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.82 - 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.32 - 9.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.878 + 1.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.07T + 83T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + (5.41 + 9.37i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.24117493356680205641772170533, −9.595639554330800134072767971918, −8.714416929705458485722814919637, −8.377826042017361462494872817514, −7.01178059864762137243654610675, −5.61063202895794141311040928795, −4.86366888583629807037465188365, −3.89486135117042234384179969046, −3.28628843028964477457574637428, −0.907469985273236493169623419300, 1.37914378363513786209924942100, 3.12648490201576576722665127460, 3.80230569223751793404263871026, 5.18026382451802604854416625686, 6.07596108670925625862274174616, 7.32472395284869883511343560244, 7.940150175434565242963216016008, 8.653480288962083679836322614591, 9.890243308521565931041170831935, 10.46942007452265331712640063243

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