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Properties

Label 2-637-91.9-c1-0-23
Degree $2$
Conductor $637$
Sign $0.270 - 0.962i$
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  + (0.651 + 1.12i)2-s + 2.88·3-s + (0.151 − 0.262i)4-s + (−1.44 + 2.49i)5-s + (1.87 + 3.25i)6-s + 3·8-s + 5.30·9-s − 3.75·10-s − 5.90·11-s + (0.436 − 0.755i)12-s + (3.31 + 1.41i)13-s + (−4.15 + 7.19i)15-s + (1.65 + 2.86i)16-s + (0.436 − 0.755i)17-s + (3.45 + 5.98i)18-s + 2.88·19-s + ⋯
L(s)  = 1  + (0.460 + 0.797i)2-s + 1.66·3-s + (0.0756 − 0.131i)4-s + (−0.644 + 1.11i)5-s + (0.766 + 1.32i)6-s + 1.06·8-s + 1.76·9-s − 1.18·10-s − 1.78·11-s + (0.125 − 0.218i)12-s + (0.920 + 0.391i)13-s + (−1.07 + 1.85i)15-s + (0.412 + 0.715i)16-s + (0.105 − 0.183i)17-s + (0.814 + 1.41i)18-s + 0.661·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.270 - 0.962i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.270 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47510 + 1.87595i\)
\(L(\frac12)\) \(\approx\) \(2.47510 + 1.87595i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.651 - 1.12i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 - 2.88T + 3T^{2} \)
5 \( 1 + (1.44 - 2.49i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 5.90T + 11T^{2} \)
17 \( 1 + (-0.436 + 0.755i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 2.88T + 19T^{2} \)
23 \( 1 + (3.30 + 5.72i)T + (-11.5 + 19.9i)T^{2} \)
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 \( 1 + (0.697 + 1.20i)T + (-18.5 + 32.0i)T^{2} \)
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 \( 1 + (3.31 - 5.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 5.76T + 61T^{2} \)
67 \( 1 - T + 67T^{2} \)
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 \( 1 + (4.32 + 7.48i)T + (-44.5 + 77.0i)T^{2} \)
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

−10.49699749711643316575148707241, −10.02969981888463258276343697200, −8.610806572793416354487099549051, −7.946483568785733280088526382491, −7.33801604818748936112154274938, −6.58307359433810253118684113289, −5.29190592980784451511227601246, −4.04365410577749753438400518967, −3.12054763696221523421398594918, −2.15573981927522526459342177190, 1.51556906462447486507820816520, 2.76829786986628938002452727435, 3.51236619596134889130721376425, 4.37406414739981818973190370366, 5.43248125701088135084387608888, 7.49996432270798622982345727291, 7.932288796893726430775260268574, 8.458174323075853464214506743472, 9.496883950739148055776416506791, 10.40652501664542633202852197284

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