Properties

Label 2-637-91.54-c1-0-11
Degree $2$
Conductor $637$
Sign $0.276 - 0.961i$
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  + (0.203 − 0.203i)2-s + (−0.923 − 0.532i)3-s + 1.91i·4-s + (−0.499 + 0.133i)5-s + (−0.296 + 0.0794i)6-s + (0.798 + 0.798i)8-s + (−0.931 − 1.61i)9-s + (−0.0744 + 0.129i)10-s + (3.69 − 0.990i)11-s + (1.02 − 1.76i)12-s + (−1.12 + 3.42i)13-s + (0.532 + 0.142i)15-s − 3.50·16-s + 4.52·17-s + (−0.518 − 0.139i)18-s + (−0.797 + 2.97i)19-s + ⋯
L(s)  = 1  + (0.144 − 0.144i)2-s + (−0.532 − 0.307i)3-s + 0.958i·4-s + (−0.223 + 0.0598i)5-s + (−0.121 + 0.0324i)6-s + (0.282 + 0.282i)8-s + (−0.310 − 0.538i)9-s + (−0.0235 + 0.0407i)10-s + (1.11 − 0.298i)11-s + (0.294 − 0.510i)12-s + (−0.312 + 0.950i)13-s + (0.137 + 0.0368i)15-s − 0.877·16-s + 1.09·17-s + (−0.122 − 0.0327i)18-s + (−0.183 + 0.683i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.276 - 0.961i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.276 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.903403 + 0.680435i\)
\(L(\frac12)\) \(\approx\) \(0.903403 + 0.680435i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.12 - 3.42i)T \)
good2 \( 1 + (-0.203 + 0.203i)T - 2iT^{2} \)
3 \( 1 + (0.923 + 0.532i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.499 - 0.133i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.69 + 0.990i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 4.52T + 17T^{2} \)
19 \( 1 + (0.797 - 2.97i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 8.67iT - 23T^{2} \)
29 \( 1 + (-1.26 - 2.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.270 - 1.00i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.129 + 0.129i)T + 37iT^{2} \)
41 \( 1 + (1.79 - 6.68i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.59 - 2.65i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.42 + 9.03i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.512 - 0.887i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.368 - 0.368i)T - 59iT^{2} \)
61 \( 1 + (-7.39 + 4.26i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.83 + 6.83i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.96 - 7.33i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-11.9 - 3.20i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.77 + 6.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.42 + 6.42i)T + 83iT^{2} \)
89 \( 1 + (8.56 - 8.56i)T - 89iT^{2} \)
97 \( 1 + (-13.1 + 3.53i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.25409553734917599462383075215, −9.746806497216331518540095658772, −9.059742946767260345168065592938, −8.027485611449246772105545660610, −7.18737882437932674554140587369, −6.37978482439057572153405544282, −5.32973316470645631692681121253, −3.90475179974052724638926491391, −3.36452710669987980317000419422, −1.56215585637019246966337931333, 0.66629104634702537423669781833, 2.40026649930492494035507270884, 4.11402716019503824849215786178, 4.95844953210645216433504890609, 5.80413392849289213868244567373, 6.57506263508144314245504085246, 7.70335553169800108423239173493, 8.730466071920706682980179021824, 9.818022536004882627018002392591, 10.37172430923070703539480195422

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