↑

Properties

Label 2-637-13.4-c1-0-3
Degree $2$
Conductor $637$
Sign $-0.0123 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.104 − 0.0601i)2-s + (−0.291 + 0.504i)3-s + (−0.992 − 1.71i)4-s + 1.68i·5-s + (0.0606 − 0.0350i)6-s + 0.479i·8-s + (1.33 + 2.30i)9-s + (0.101 − 0.175i)10-s + (−0.315 − 0.182i)11-s + 1.15·12-s + (−1.80 − 3.12i)13-s + (−0.851 − 0.491i)15-s + (−1.95 + 3.38i)16-s + (1.59 + 2.75i)17-s − 0.320i·18-s + (−1.25 + 0.721i)19-s + ⋯
L(s)  = 1  + (−0.0737 − 0.0425i)2-s + (−0.168 + 0.291i)3-s + (−0.496 − 0.859i)4-s + 0.754i·5-s + (0.0247 − 0.0143i)6-s + 0.169i·8-s + (0.443 + 0.768i)9-s + (0.0321 − 0.0556i)10-s + (−0.0952 − 0.0549i)11-s + 0.333·12-s + (−0.499 − 0.866i)13-s + (−0.219 − 0.126i)15-s + (−0.489 + 0.847i)16-s + (0.386 + 0.669i)17-s − 0.0754i·18-s + (−0.286 + 0.165i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0123 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0123 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.0123 - 0.999i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.0123 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.644583 + 0.652582i\)
\(L(\frac12)\) \(\approx\) \(0.644583 + 0.652582i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 + 3.12i)T \)
good2 \( 1 + (0.104 + 0.0601i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.291 - 0.504i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.68iT - 5T^{2} \)
11 \( 1 + (0.315 + 0.182i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.59 - 2.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.25 - 0.721i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.54 - 4.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.09 - 7.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.69iT - 31T^{2} \)
37 \( 1 + (-5.46 - 3.15i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.04 - 2.91i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.386 + 0.669i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.7iT - 47T^{2} \)
53 \( 1 - 1.37T + 53T^{2} \)
59 \( 1 + (-8.10 + 4.68i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.51 + 7.81i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.6 + 6.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.13 - 3.54i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.16iT - 73T^{2} \)
79 \( 1 + 6.88T + 79T^{2} \)
83 \( 1 + 0.567iT - 83T^{2} \)
89 \( 1 + (-0.986 - 0.569i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.86 - 3.96i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.64402092615038832419666022398, −10.12180230892638949410210512228, −9.329470002753793056186116334566, −8.137521648754793163530841588893, −7.32022819236400873939993556050, −6.12392792298183068401409390365, −5.34182945774415193601184954100, −4.43843649659707487018885059313, −3.10842799311813917497012029221, −1.59863891393254811190849073567, 0.54545235710574086588830763407, 2.40829896939217048616886759910, 3.98516923201743405983421117466, 4.54294643658024321013022116799, 5.84001989974515358004744574212, 7.01313425193476714245183458323, 7.67210098967122421625082464037, 8.752624095033744793276273327923, 9.299566630730084316509597046271, 10.11205227579103668657584674099

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