↑

Properties

Label 2-637-91.51-c1-0-34
Degree $2$
Conductor $637$
Sign $-0.00450 + 0.999i$
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  + (1.01 + 0.585i)2-s + (−0.269 − 0.466i)3-s + (−0.315 − 0.546i)4-s + (−0.399 − 0.230i)5-s − 0.630i·6-s − 3.07i·8-s + (1.35 − 2.34i)9-s + (−0.269 − 0.466i)10-s + (−0.718 + 0.414i)11-s + (−0.170 + 0.294i)12-s + (−2.87 + 2.17i)13-s + 0.248i·15-s + (1.17 − 2.02i)16-s + (−1.43 − 2.49i)17-s + (2.74 − 1.58i)18-s + (−3.74 − 2.16i)19-s + ⋯
L(s)  = 1  + (0.716 + 0.413i)2-s + (−0.155 − 0.269i)3-s + (−0.157 − 0.273i)4-s + (−0.178 − 0.103i)5-s − 0.257i·6-s − 1.08i·8-s + (0.451 − 0.782i)9-s + (−0.0852 − 0.147i)10-s + (−0.216 + 0.125i)11-s + (−0.0490 + 0.0850i)12-s + (−0.798 + 0.601i)13-s + 0.0641i·15-s + (0.292 − 0.506i)16-s + (−0.349 − 0.604i)17-s + (0.647 − 0.373i)18-s + (−0.859 − 0.496i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00450 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00450 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.00450 + 0.999i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.00450 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05222 - 1.05697i\)
\(L(\frac12)\) \(\approx\) \(1.05222 - 1.05697i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.87 - 2.17i)T \)
good2 \( 1 + (-1.01 - 0.585i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.269 + 0.466i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.399 + 0.230i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.718 - 0.414i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.43 + 2.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.74 + 2.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.52 + 4.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.261T + 29T^{2} \)
31 \( 1 + (-5.88 + 3.40i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.23 - 4.75i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.68iT - 41T^{2} \)
43 \( 1 - 0.418T + 43T^{2} \)
47 \( 1 + (-8.00 - 4.62i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.815 + 1.41i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.41 + 1.39i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.63 - 6.28i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.39 - 2.53i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + (0.306 - 0.176i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.40 - 2.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + (-4.70 - 2.71i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.6iT - 97T^{2} \)
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

−10.22387094905328011994308748094, −9.567572943107869143684102424345, −8.664935752946966773591304574733, −7.32969523281987091021678122415, −6.67062888722310186524473561463, −5.92592255055626000268783384276, −4.59823519765407307829600890991, −4.24891656102547978090640997196, −2.54020560688678281431731453621, −0.65593011318475456156890374570, 2.08820828818567189186087281593, 3.25208061125664591639975938831, 4.30781123122549782493721188464, 5.02643634553952025278001106699, 5.99536851370409889277558421392, 7.50092329474975157640863051824, 8.000333298258375774362480175027, 9.105548193521683693894367820649, 10.21770272879547650104461091133, 10.88087819859763275482547403224

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