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Properties

Label 2-637-91.81-c1-0-42
Degree $2$
Conductor $637$
Sign $-0.113 - 0.993i$
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  + (1.20 − 2.09i)2-s − 1.41·3-s + (−1.91 − 3.31i)4-s + (−1.91 − 3.31i)5-s + (−1.70 + 2.95i)6-s − 4.41·8-s − 0.999·9-s − 9.24·10-s + 3.41·11-s + (2.70 + 4.68i)12-s + (−3.5 + 0.866i)13-s + (2.70 + 4.68i)15-s + (−1.49 + 2.59i)16-s + (0.0857 + 0.148i)17-s + (−1.20 + 2.09i)18-s + 6·19-s + ⋯
L(s)  = 1  + (0.853 − 1.47i)2-s − 0.816·3-s + (−0.957 − 1.65i)4-s + (−0.856 − 1.48i)5-s + (−0.696 + 1.20i)6-s − 1.56·8-s − 0.333·9-s − 2.92·10-s + 1.02·11-s + (0.781 + 1.35i)12-s + (−0.970 + 0.240i)13-s + (0.698 + 1.21i)15-s + (−0.374 + 0.649i)16-s + (0.0208 + 0.0360i)17-s + (−0.284 + 0.492i)18-s + 1.37·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.113 - 0.993i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.113 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.593786 + 0.665603i\)
\(L(\frac12)\) \(\approx\) \(0.593786 + 0.665603i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 + 2.09i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 1.41T + 3T^{2} \)
5 \( 1 + (1.91 + 3.31i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 3.41T + 11T^{2} \)
17 \( 1 + (-0.0857 - 0.148i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + (0.707 - 1.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.91 + 8.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.70 - 4.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.74 - 6.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.91 + 5.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.292 - 0.507i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.82 + 6.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.878 - 1.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 9.82T + 61T^{2} \)
67 \( 1 + 4.24T + 67T^{2} \)
71 \( 1 + (-0.171 + 0.297i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.328 - 0.568i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.12 + 8.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 + (-3.65 + 6.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.58 + 4.47i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.12019836406894005073649067390, −9.396326286772014309728901758357, −8.500280742828886589662518294693, −7.20816186267345972466884967230, −5.65163846336988181151201707159, −5.06041902576579618316353196856, −4.28260351428739860819926292435, −3.35898316871274699661793414253, −1.62759586374102032425066402327, −0.41905424541474725897567591108, 3.08144851496410762630810795252, 3.92971702860887586271467173199, 5.11977815984498563838506659718, 5.89272355207702351268705846897, 6.86209090588049083146662626683, 7.19505016904438601886433281364, 8.052253923369356853386892702374, 9.327481877734618195698970770203, 10.58933809490194377186453139362, 11.44287059883156436550360901508

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