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Properties

Label 2-637-91.74-c1-0-8
Degree $2$
Conductor $637$
Sign $0.166 - 0.986i$
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  − 0.851·2-s + (0.330 + 0.572i)3-s − 1.27·4-s + (1.72 + 2.98i)5-s + (−0.281 − 0.487i)6-s + 2.78·8-s + (1.28 − 2.21i)9-s + (−1.46 − 2.53i)10-s + (0.448 + 0.777i)11-s + (−0.421 − 0.730i)12-s + (3.07 + 1.88i)13-s + (−1.13 + 1.97i)15-s + 0.178·16-s − 1.93·17-s + (−1.09 + 1.88i)18-s + (0.519 − 0.898i)19-s + ⋯
L(s)  = 1  − 0.601·2-s + (0.190 + 0.330i)3-s − 0.637·4-s + (0.769 + 1.33i)5-s + (−0.114 − 0.198i)6-s + 0.985·8-s + (0.427 − 0.739i)9-s + (−0.463 − 0.802i)10-s + (0.135 + 0.234i)11-s + (−0.121 − 0.210i)12-s + (0.852 + 0.522i)13-s + (−0.293 + 0.508i)15-s + 0.0445·16-s − 0.469·17-s + (−0.257 + 0.445i)18-s + (0.119 − 0.206i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.166 - 0.986i$
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: \((2,\ 637,\ (\ :1/2),\ 0.166 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904634 + 0.764791i\)
\(L(\frac12)\) \(\approx\) \(0.904634 + 0.764791i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.07 - 1.88i)T \)
good2 \( 1 + 0.851T + 2T^{2} \)
3 \( 1 + (-0.330 - 0.572i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.72 - 2.98i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.448 - 0.777i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.93T + 17T^{2} \)
19 \( 1 + (-0.519 + 0.898i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 + (-0.917 + 1.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.56 - 7.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 + (2.66 - 4.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.95 - 3.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.59 - 6.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.69 + 8.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.510T + 59T^{2} \)
61 \( 1 + (-0.718 + 1.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.22 - 7.31i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.72 - 2.98i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.45 + 9.44i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.04 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.51T + 83T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + (-0.253 - 0.438i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.60390111271304028177369479272, −9.799584181759390403597692670364, −9.197173798791232629151121547467, −8.474659867669004016035566575084, −6.99382723737369143806228873288, −6.68864809758227853995763730153, −5.30402111036723151016176378391, −4.07358974714031921791306053081, −3.10619359074108145553187176376, −1.51937051905495275059087021785, 0.909927814578637257170161400004, 1.93368373237992002653196236689, 3.88484061209091821009391862634, 5.01518241463518524145707078929, 5.59334580589867659421277680375, 7.07117368700537698717166491518, 8.085155620005702383443219403943, 8.788814432702224692554373137207, 9.204998993445784675646312369508, 10.26737161043478935141636779891

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