Properties

Label 2-690-5.3-c2-0-27
Degree $2$
Conductor $690$
Sign $0.997 + 0.0663i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
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 + i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + (4.77 + 1.46i)5-s + 2.44·6-s + (8.57 − 8.57i)7-s + (2 + 2i)8-s + 2.99i·9-s + (−6.24 + 3.31i)10-s + 5.33·11-s + (−2.44 + 2.44i)12-s + (9.44 + 9.44i)13-s + 17.1i·14-s + (−4.05 − 7.65i)15-s − 4·16-s + (15.8 − 15.8i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (0.955 + 0.293i)5-s + 0.408·6-s + (1.22 − 1.22i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.624 + 0.331i)10-s + 0.485·11-s + (−0.204 + 0.204i)12-s + (0.726 + 0.726i)13-s + 1.22i·14-s + (−0.270 − 0.510i)15-s − 0.250·16-s + (0.932 − 0.932i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0663i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.997 + 0.0663i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.997 + 0.0663i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.859881992\)
\(L(\frac12)\) \(\approx\) \(1.859881992\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (-4.77 - 1.46i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good7 \( 1 + (-8.57 + 8.57i)T - 49iT^{2} \)
11 \( 1 - 5.33T + 121T^{2} \)
13 \( 1 + (-9.44 - 9.44i)T + 169iT^{2} \)
17 \( 1 + (-15.8 + 15.8i)T - 289iT^{2} \)
19 \( 1 - 27.9iT - 361T^{2} \)
29 \( 1 + 43.2iT - 841T^{2} \)
31 \( 1 + 45.6T + 961T^{2} \)
37 \( 1 + (22.6 - 22.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 27.4T + 1.68e3T^{2} \)
43 \( 1 + (2.61 + 2.61i)T + 1.84e3iT^{2} \)
47 \( 1 + (60.8 - 60.8i)T - 2.20e3iT^{2} \)
53 \( 1 + (-9.28 - 9.28i)T + 2.80e3iT^{2} \)
59 \( 1 + 62.7iT - 3.48e3T^{2} \)
61 \( 1 - 100.T + 3.72e3T^{2} \)
67 \( 1 + (23.2 - 23.2i)T - 4.48e3iT^{2} \)
71 \( 1 - 73.4T + 5.04e3T^{2} \)
73 \( 1 + (-74.1 - 74.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 42.9iT - 6.24e3T^{2} \)
83 \( 1 + (-30.3 - 30.3i)T + 6.88e3iT^{2} \)
89 \( 1 + 115. iT - 7.92e3T^{2} \)
97 \( 1 + (-131. + 131. i)T - 9.40e3iT^{2} \)
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

−10.13268139619141479953177140129, −9.548786367879561751008427819559, −8.309983718153122528731061106333, −7.57606180859814162934894536380, −6.80797317574864978613424934614, −5.93390792136911950853580026171, −5.06028134545651523025438106183, −3.84326762917728451924496609643, −1.82084327291995461890107083457, −1.09904163002360249190822266780, 1.18282651698702998834083362361, 2.18423133442150373829198538378, 3.55236239278684456512016198726, 5.09904915575077389872962288526, 5.47072962425881436124034488010, 6.62712512250163175555815407827, 8.031449099429020454227414446249, 8.914203615350314075938287947004, 9.191146696671191552828476474308, 10.47689940444351585928314793866

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