↑

Properties

Label 2-637-91.23-c1-0-38
Degree $2$
Conductor $637$
Sign $-0.964 + 0.263i$
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  − 1.38i·2-s + (1.41 − 2.44i)3-s + 0.0791·4-s + (0.449 + 0.259i)5-s + (−3.39 − 1.95i)6-s − 2.88i·8-s + (−2.49 − 4.31i)9-s + (0.359 − 0.622i)10-s + (−1.40 − 0.812i)11-s + (0.111 − 0.193i)12-s + (1.42 + 3.31i)13-s + (1.26 − 0.733i)15-s − 3.83·16-s − 1.94·17-s + (−5.98 + 3.45i)18-s + (−2.15 + 1.24i)19-s + ⋯
L(s)  = 1  − 0.980i·2-s + (0.815 − 1.41i)3-s + 0.0395·4-s + (0.200 + 0.116i)5-s + (−1.38 − 0.799i)6-s − 1.01i·8-s + (−0.830 − 1.43i)9-s + (0.113 − 0.196i)10-s + (−0.424 − 0.244i)11-s + (0.0322 − 0.0559i)12-s + (0.395 + 0.918i)13-s + (0.327 − 0.189i)15-s − 0.958·16-s − 0.472·17-s + (−1.41 + 0.814i)18-s + (−0.494 + 0.285i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.964 + 0.263i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.964 + 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.285845 - 2.12710i\)
\(L(\frac12)\) \(\approx\) \(0.285845 - 2.12710i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 - 3.31i)T \)
good2 \( 1 + 1.38iT - 2T^{2} \)
3 \( 1 + (-1.41 + 2.44i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.449 - 0.259i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.40 + 0.812i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.94T + 17T^{2} \)
19 \( 1 + (2.15 - 1.24i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 9.14T + 23T^{2} \)
29 \( 1 + (-2.61 - 4.52i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.01 - 2.89i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 + (-3.64 + 2.10i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.498 - 0.863i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.91 - 2.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.44 - 7.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.20iT - 59T^{2} \)
61 \( 1 + (-6.73 - 11.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.25 - 4.18i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.50 + 2.59i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (10.2 - 5.91i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.491 - 0.850i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.91iT - 83T^{2} \)
89 \( 1 + 12.0iT - 89T^{2} \)
97 \( 1 + (-3.82 - 2.21i)T + (48.5 + 84.0i)T^{2} \)
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

−10.44787636066128208205353356021, −9.077438245078737323311885214584, −8.702961328369168420874837544886, −7.30228162294964639463400837271, −6.94316889882362163881654057156, −5.89420040353935538341907445209, −4.10247615975847185826337187328, −2.89910144095185311192409752383, −2.18555782872539672204316516091, −1.13272787588877137482176456384, 2.42198744320962816878833523516, 3.44082843879742382614576330610, 4.76907556040422341585964297121, 5.34537495632775056383939428992, 6.51105157422266154189220429184, 7.62725792698524095111406901021, 8.409783446461402680999457262637, 9.069362792525211195779345996906, 9.962371294492806244931109453604, 10.81194916166591384677444889891

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