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Properties

Label 2-637-7.4-c1-0-8
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  + (0.707 + 1.22i)2-s + (−0.707 + 1.22i)3-s + (−0.792 − 1.37i)5-s − 2·6-s + 2.82·8-s + (0.500 + 0.866i)9-s + (1.12 − 1.94i)10-s + (−2.12 + 3.67i)11-s − 13-s + 2.24·15-s + (2.00 + 3.46i)16-s + (−0.707 + 1.22i)17-s + (−0.707 + 1.22i)18-s + (3.62 + 6.27i)19-s − 6·22-s + (2.91 + 5.04i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s + (−0.408 + 0.707i)3-s + (−0.354 − 0.614i)5-s − 0.816·6-s + 0.999·8-s + (0.166 + 0.288i)9-s + (0.354 − 0.614i)10-s + (−0.639 + 1.10i)11-s − 0.277·13-s + 0.579·15-s + (0.500 + 0.866i)16-s + (−0.171 + 0.297i)17-s + (−0.166 + 0.288i)18-s + (0.830 + 1.43i)19-s − 1.27·22-s + (0.607 + 1.05i)23-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: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.729232 + 1.47103i\)
\(L(\frac12)\) \(\approx\) \(0.729232 + 1.47103i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 1.22i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.707 - 1.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.792 + 1.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.12 - 3.67i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.707 - 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.62 - 6.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.91 - 5.04i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.171T + 29T^{2} \)
31 \( 1 + (1.62 - 2.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.12 + 1.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.82T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (0.792 + 1.37i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0857 + 0.148i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.171 - 0.297i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.24 + 12.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 + (-4.62 + 8.00i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.74 + 13.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (0.792 + 1.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−10.64992905560698935619700396110, −10.16252421522331386812271899411, −9.249022963004694096332893213636, −7.78062353921991098753323442338, −7.52155129495034149648520088515, −6.20273345457375572043241372900, −5.14020520497608031039897388308, −4.85699648190543861347734672917, −3.79209419023328486459927886650, −1.75582133398507214641648096402, 0.843035439869330365332228107079, 2.57718109392760885401424828131, 3.25519199638047478425067199214, 4.51526380296329360908899372453, 5.65125628102944348544439401092, 6.93834892555442160158262301177, 7.30138030968960101635145609711, 8.438587336969318124915373250078, 9.637639951544146917254813861506, 10.81005655927754517179494615289

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