↑

Properties

Label 2-637-91.88-c1-0-32
Degree $2$
Conductor $637$
Sign $0.268 - 0.963i$
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  + (2.34 + 1.35i)2-s + 0.345·3-s + (2.65 + 4.59i)4-s + (2.82 − 1.62i)5-s + (0.809 + 0.467i)6-s + 8.94i·8-s − 2.88·9-s + 8.80·10-s − 1.84i·11-s + (0.918 + 1.59i)12-s + (−3.60 − 0.0186i)13-s + (0.976 − 0.563i)15-s + (−6.77 + 11.7i)16-s + (−1.07 − 1.86i)17-s + (−6.74 − 3.89i)18-s − 2.40i·19-s + ⋯
L(s)  = 1  + (1.65 + 0.955i)2-s + 0.199·3-s + (1.32 + 2.29i)4-s + (1.26 − 0.728i)5-s + (0.330 + 0.190i)6-s + 3.16i·8-s − 0.960·9-s + 2.78·10-s − 0.556i·11-s + (0.265 + 0.459i)12-s + (−0.999 − 0.00517i)13-s + (0.252 − 0.145i)15-s + (−1.69 + 2.93i)16-s + (−0.261 − 0.452i)17-s + (−1.58 − 0.917i)18-s − 0.550i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.268 - 0.963i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.268 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.43116 + 2.60649i\)
\(L(\frac12)\) \(\approx\) \(3.43116 + 2.60649i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.60 + 0.0186i)T \)
good2 \( 1 + (-2.34 - 1.35i)T + (1 + 1.73i)T^{2} \)
3 \( 1 - 0.345T + 3T^{2} \)
5 \( 1 + (-2.82 + 1.62i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 1.84iT - 11T^{2} \)
17 \( 1 + (1.07 + 1.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 2.40iT - 19T^{2} \)
23 \( 1 + (-0.906 + 1.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.36 - 2.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.50 + 0.871i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.14 - 2.96i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.65 - 2.11i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.34 - 7.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.09 - 2.93i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.65 + 8.05i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.31 + 5.37i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 - 0.826iT - 67T^{2} \)
71 \( 1 + (2.03 + 1.17i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.76 - 1.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.400 + 0.694i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.97iT - 83T^{2} \)
89 \( 1 + (-13.0 - 7.55i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.99 + 4.61i)T + (48.5 + 84.0i)T^{2} \)
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

−11.17990674060128315395887014445, −9.705624855043814991473938211356, −8.756697810225026531429147815428, −7.980075479978405013757909873724, −6.78648108310372167688527395329, −6.06661285465989164940075219887, −5.20572546400712146658191079865, −4.74026321616456379325352147140, −3.18177235196719829161518767670, −2.33418749630629203763372828522, 1.94530474965301537952856078554, 2.54679386008515733493008014238, 3.56158159836658934651868752640, 4.84235353995365392565847309289, 5.69365072544448575164601746323, 6.31121105631474630259478728264, 7.32878259242982791798032158950, 9.089063947660372989185230262030, 10.09924961410608758021906883115, 10.40031564754274049743687623838

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