↑

Properties

Label 2-637-91.4-c1-0-18
Degree $2$
Conductor $637$
Sign $0.958 - 0.284i$
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  + 2.70i·2-s + (−0.172 − 0.299i)3-s − 5.30·4-s + (−2.82 + 1.62i)5-s + (0.809 − 0.467i)6-s − 8.94i·8-s + (1.44 − 2.49i)9-s + (−4.40 − 7.62i)10-s + (1.59 − 0.923i)11-s + (0.918 + 1.59i)12-s + (−3.60 + 0.0186i)13-s + (0.976 + 0.563i)15-s + 13.5·16-s + 2.15·17-s + (6.74 + 3.89i)18-s + (−2.07 − 1.20i)19-s + ⋯
L(s)  = 1  + 1.91i·2-s + (−0.0998 − 0.172i)3-s − 2.65·4-s + (−1.26 + 0.728i)5-s + (0.330 − 0.190i)6-s − 3.16i·8-s + (0.480 − 0.831i)9-s + (−1.39 − 2.41i)10-s + (0.482 − 0.278i)11-s + (0.265 + 0.459i)12-s + (−0.999 + 0.00517i)13-s + (0.252 + 0.145i)15-s + 3.38·16-s + 0.522·17-s + (1.58 + 0.917i)18-s + (−0.477 − 0.275i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.958 - 0.284i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.958 - 0.284i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.447462 + 0.0649049i\)
\(L(\frac12)\) \(\approx\) \(0.447462 + 0.0649049i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.70iT - 2T^{2} \)
3 \( 1 + (0.172 + 0.299i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.82 - 1.62i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.59 + 0.923i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 2.15T + 17T^{2} \)
19 \( 1 + (2.07 + 1.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.81T + 23T^{2} \)
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 \( 1 - 5.93iT - 37T^{2} \)
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 \( 1 + 10.7iT - 59T^{2} \)
61 \( 1 + (-5.05 + 8.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.716 - 0.413i)T + (33.5 - 58.0i)T^{2} \)
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 \( 1 - 15.1iT - 89T^{2} \)
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

−10.28299622335983204329459948744, −9.477476023534748693060755850294, −8.424865743412350333533071461288, −7.79796847469254708414456585902, −6.81736095530045666553918863578, −6.69090770984146821374830772271, −5.34512984709714351816606182337, −4.24432498268471682776022512022, −3.50440857389042754915765040847, −0.28659544109106217206245615921, 1.30886880264668729641349443231, 2.65715827526571023779244571872, 4.02785830595056904723255492251, 4.41906881978469957776561215445, 5.33715824609398848211000944358, 7.44715122740996689907934087470, 8.217660240091854094218143219484, 9.078669026412622672100513019553, 9.972786687173490394624331512084, 10.59536640429521489980698856026

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