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Properties

Label 2-637-91.81-c1-0-3
Degree $2$
Conductor $637$
Sign $-0.868 + 0.495i$
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  + (−0.5 + 0.866i)2-s − 1.41·3-s + (0.500 + 0.866i)4-s + (2.04 + 3.54i)5-s + (0.707 − 1.22i)6-s − 3·8-s − 0.999·9-s − 4.09·10-s − 3.79·11-s + (−0.707 − 1.22i)12-s + (0.634 + 3.54i)13-s + (−2.89 − 5.01i)15-s + (0.500 − 0.866i)16-s + (−0.634 − 1.09i)17-s + (0.499 − 0.866i)18-s + 2.82·19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s − 0.816·3-s + (0.250 + 0.433i)4-s + (0.916 + 1.58i)5-s + (0.288 − 0.499i)6-s − 1.06·8-s − 0.333·9-s − 1.29·10-s − 1.14·11-s + (−0.204 − 0.353i)12-s + (0.176 + 0.984i)13-s + (−0.748 − 1.29i)15-s + (0.125 − 0.216i)16-s + (−0.153 − 0.266i)17-s + (0.117 − 0.204i)18-s + 0.648·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.868 + 0.495i$
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: \((2,\ 637,\ (\ :1/2),\ -0.868 + 0.495i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.184238 - 0.695569i\)
\(L(\frac12)\) \(\approx\) \(0.184238 - 0.695569i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.634 - 3.54i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 1.41T + 3T^{2} \)
5 \( 1 + (-2.04 - 3.54i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
17 \( 1 + (0.634 + 1.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + (-3.89 + 6.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.397 + 0.689i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.707 - 1.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.39 - 2.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.48 - 2.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.89 - 6.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.41 - 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.29 + 10.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.21 + 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 8.34T + 61T^{2} \)
67 \( 1 + 3.79T + 67T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.29 - 10.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.10 - 1.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + (7.48 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.12 - 3.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.10064348696285968060133140065, −10.32927784575772438871216721040, −9.402926786429349788734500216737, −8.331717815048563281059701864542, −7.24950707947736524559869039271, −6.60958331561683117131520109307, −6.07049159368413777380266345123, −5.04053182571375772429004617306, −3.15597723643001390003042288695, −2.43659558683886248675182424817, 0.46682970441815811939439641500, 1.58568920275355447430556738076, 2.95317863516787812009234013017, 4.91596141733055030950748950112, 5.65763217936776044982233883589, 5.82681366490465296675238304160, 7.52448590049559961057111767302, 8.740673638834760974537849374333, 9.257325698074129451364745760988, 10.32385351185034509177486697081

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