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Properties

Label 2-637-7.2-c1-0-20
Degree $2$
Conductor $637$
Sign $0.605 - 0.795i$
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  + (−1.17 + 2.02i)2-s + (0.573 + 0.992i)3-s + (−1.74 − 3.02i)4-s + (0.671 − 1.16i)5-s − 2.68·6-s + 3.48·8-s + (0.842 − 1.45i)9-s + (1.57 + 2.72i)10-s + (−0.573 − 0.992i)11-s + (2 − 3.46i)12-s + 13-s + 1.53·15-s + (−0.598 + 1.03i)16-s + (−2.91 − 5.05i)17-s + (1.97 + 3.42i)18-s + (1.67 − 2.89i)19-s + ⋯
L(s)  = 1  + (−0.828 + 1.43i)2-s + (0.330 + 0.573i)3-s + (−0.872 − 1.51i)4-s + (0.300 − 0.520i)5-s − 1.09·6-s + 1.23·8-s + (0.280 − 0.486i)9-s + (0.497 + 0.861i)10-s + (−0.172 − 0.299i)11-s + (0.577 − 0.999i)12-s + 0.277·13-s + 0.397·15-s + (−0.149 + 0.259i)16-s + (−0.707 − 1.22i)17-s + (0.465 + 0.806i)18-s + (0.383 − 0.664i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.941538 + 0.466746i\)
\(L(\frac12)\) \(\approx\) \(0.941538 + 0.466746i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + (1.17 - 2.02i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.573 - 0.992i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.671 + 1.16i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.573 + 0.992i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.91 + 5.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.67 + 2.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.58 + 2.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
31 \( 1 + (0.817 + 1.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.25 - 7.37i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.292T + 41T^{2} \)
43 \( 1 + 8.15T + 43T^{2} \)
47 \( 1 + (-5.30 + 9.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.391 - 0.677i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.32 + 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.05 - 5.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.53T + 71T^{2} \)
73 \( 1 + (-7.65 - 13.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.441 - 0.764i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + (2.86 - 4.96i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.34T + 97T^{2} \)
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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

−10.12175947129014270040579294351, −9.588103372721924940879297306073, −8.720092581491891834962973391074, −8.436893006677318981978114781318, −7.00114149927471537423227586042, −6.60783271472176063174175302832, −5.27376040689103822706882851662, −4.66502741272592187581587477565, −3.04271063713227250216556200830, −0.821294400927533823881995176095, 1.40046142178110745962061108358, 2.27793397399067280805995364883, 3.25573402263557505245686635144, 4.49953796443089226654373359208, 6.12026712749926761754033053875, 7.19156332721071557759182944943, 8.180195219483340050366861115161, 8.756580632829602207354766995578, 9.866063339378579790894242287643, 10.54010086325438131626714470570

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