↑

Properties

Label 2-637-91.25-c1-0-3
Degree $2$
Conductor $637$
Sign $-0.637 + 0.770i$
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  + (−0.596 + 0.344i)2-s + (−1.10 + 1.91i)3-s + (−0.762 + 1.32i)4-s + (2.78 − 1.60i)5-s − 1.52i·6-s − 2.42i·8-s + (−0.951 − 1.64i)9-s + (−1.10 + 1.91i)10-s + (−2.32 − 1.34i)11-s + (−1.68 − 2.92i)12-s + (−3.59 + 0.311i)13-s + 7.11i·15-s + (−0.688 − 1.19i)16-s + (−1.79 + 3.11i)17-s + (1.13 + 0.655i)18-s + (−7.40 + 4.27i)19-s + ⋯
L(s)  = 1  + (−0.421 + 0.243i)2-s + (−0.639 + 1.10i)3-s + (−0.381 + 0.660i)4-s + (1.24 − 0.718i)5-s − 0.622i·6-s − 0.858i·8-s + (−0.317 − 0.549i)9-s + (−0.350 + 0.606i)10-s + (−0.702 − 0.405i)11-s + (−0.487 − 0.844i)12-s + (−0.996 + 0.0862i)13-s + 1.83i·15-s + (−0.172 − 0.298i)16-s + (−0.435 + 0.754i)17-s + (0.267 + 0.154i)18-s + (−1.69 + 0.980i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.637 + 0.770i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.637 + 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0897209 - 0.190556i\)
\(L(\frac12)\) \(\approx\) \(0.0897209 - 0.190556i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.59 - 0.311i)T \)
good2 \( 1 + (0.596 - 0.344i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.10 - 1.91i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.78 + 1.60i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.32 + 1.34i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.79 - 3.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.40 - 4.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.64 + 2.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.05T + 29T^{2} \)
31 \( 1 + (5.05 + 2.91i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.40 + 1.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.755iT - 41T^{2} \)
43 \( 1 + 8.80T + 43T^{2} \)
47 \( 1 + (1.63 - 0.941i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.26 - 2.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.34 - 3.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.52 - 7.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.371 - 0.214i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.98iT - 71T^{2} \)
73 \( 1 + (-5.01 - 2.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.23 - 3.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.8iT - 83T^{2} \)
89 \( 1 + (4.64 - 2.68i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.62iT - 97T^{2} \)
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

−10.67270093182146612038228656369, −10.17356379584604951674060764972, −9.492427589970393619198180704796, −8.683975800933566005368477590744, −7.933594120619769464117141111272, −6.47506863393120947830083672654, −5.60830387224371992859847411576, −4.72374084085365749845141971991, −3.95311921759675327122097631940, −2.21289330996749785994757057867, 0.13151909560832977634302917547, 1.86340722553436538004703768023, 2.46640719856414320010077662082, 4.83859754026413642040706800413, 5.57707069564988358347241657356, 6.56426872516048686274821975782, 7.03797444392413115385616279610, 8.299839942276452821953055209097, 9.480409586047019130209413020898, 9.995989794623179788874264153059

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