Properties

Label 2-637-13.3-c1-0-10
Degree $2$
Conductor $637$
Sign $-0.0662 - 0.997i$
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  + (0.952 + 1.65i)2-s + (−0.214 − 0.371i)3-s + (−0.815 + 1.41i)4-s − 1.47·5-s + (0.408 − 0.707i)6-s + 0.702·8-s + (1.40 − 2.43i)9-s + (−1.40 − 2.43i)10-s + (2.19 + 3.80i)11-s + 0.698·12-s + (2.69 + 2.39i)13-s + (0.315 + 0.546i)15-s + (2.30 + 3.98i)16-s + (0.601 − 1.04i)17-s + 5.36·18-s + (−1.62 + 2.80i)19-s + ⋯
L(s)  = 1  + (0.673 + 1.16i)2-s + (−0.123 − 0.214i)3-s + (−0.407 + 0.706i)4-s − 0.658·5-s + (0.166 − 0.288i)6-s + 0.248·8-s + (0.469 − 0.813i)9-s + (−0.443 − 0.768i)10-s + (0.662 + 1.14i)11-s + 0.201·12-s + (0.748 + 0.663i)13-s + (0.0814 + 0.141i)15-s + (0.575 + 0.996i)16-s + (0.145 − 0.252i)17-s + 1.26·18-s + (−0.371 + 0.644i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0662 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0662 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.0662 - 0.997i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.0662 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40909 + 1.50575i\)
\(L(\frac12)\) \(\approx\) \(1.40909 + 1.50575i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.69 - 2.39i)T \)
good2 \( 1 + (-0.952 - 1.65i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.214 + 0.371i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.47T + 5T^{2} \)
11 \( 1 + (-2.19 - 3.80i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.601 + 1.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.62 - 2.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.21 - 3.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0837 + 0.145i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.24T + 31T^{2} \)
37 \( 1 + (3.52 + 6.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.58 + 4.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0113 - 0.0197i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 0.141T + 53T^{2} \)
59 \( 1 + (-2.67 + 4.62i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.77 - 9.99i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.06 + 3.58i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.98 + 8.63i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 - 0.774T + 79T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 + (3.27 + 5.67i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.74 - 3.02i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.90702266304365590325204561056, −9.787824294091729799243845524717, −8.887781003680097860608983447907, −7.75982460978844246548942432655, −7.07303083264631803392344912710, −6.48032133730041205008355964857, −5.50844250991092257129829268789, −4.21920701979160949492752006898, −3.84455801116207752290261234343, −1.57132682732571764607067727754, 1.11186390317145017501669058865, 2.68869827197297701579190980266, 3.68811983037116260874689024031, 4.41789106986790159110214975531, 5.44531496134691402033899011491, 6.66472314872744668763176406787, 7.921916494084581499168982952774, 8.577847487913876485530147601375, 9.905589574125748778317049366988, 10.78163355720228341724610990706

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