Properties

Label 2-690-5.3-c2-0-14
Degree $2$
Conductor $690$
Sign $-0.238 - 0.971i$
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 + (0.0428 + 4.99i)5-s + 2.44·6-s + (0.599 − 0.599i)7-s + (2 + 2i)8-s + 2.99i·9-s + (−5.04 − 4.95i)10-s + 2.79·11-s + (−2.44 + 2.44i)12-s + (10.4 + 10.4i)13-s + 1.19i·14-s + (6.07 − 6.17i)15-s − 4·16-s + (8.12 − 8.12i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (0.00856 + 0.999i)5-s + 0.408·6-s + (0.0856 − 0.0856i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.504 − 0.495i)10-s + 0.253·11-s + (−0.204 + 0.204i)12-s + (0.801 + 0.801i)13-s + 0.0856i·14-s + (0.404 − 0.411i)15-s − 0.250·16-s + (0.477 − 0.477i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.238 - 0.971i$
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: \((2,\ 690,\ (\ :1),\ -0.238 - 0.971i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.062291605\)
\(L(\frac12)\) \(\approx\) \(1.062291605\)
\(L(2)\) 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 \( 1 + (-0.0428 - 4.99i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good7 \( 1 + (-0.599 + 0.599i)T - 49iT^{2} \)
11 \( 1 - 2.79T + 121T^{2} \)
13 \( 1 + (-10.4 - 10.4i)T + 169iT^{2} \)
17 \( 1 + (-8.12 + 8.12i)T - 289iT^{2} \)
19 \( 1 + 19.7iT - 361T^{2} \)
29 \( 1 - 53.8iT - 841T^{2} \)
31 \( 1 + 24.2T + 961T^{2} \)
37 \( 1 + (-21.3 + 21.3i)T - 1.36e3iT^{2} \)
41 \( 1 - 68.1T + 1.68e3T^{2} \)
43 \( 1 + (16.0 + 16.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (10.0 - 10.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (-20.3 - 20.3i)T + 2.80e3iT^{2} \)
59 \( 1 - 113. iT - 3.48e3T^{2} \)
61 \( 1 + 70.3T + 3.72e3T^{2} \)
67 \( 1 + (40.1 - 40.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 117.T + 5.04e3T^{2} \)
73 \( 1 + (-75.3 - 75.3i)T + 5.32e3iT^{2} \)
79 \( 1 - 84.1iT - 6.24e3T^{2} \)
83 \( 1 + (78.5 + 78.5i)T + 6.88e3iT^{2} \)
89 \( 1 - 67.9iT - 7.92e3T^{2} \)
97 \( 1 + (74.6 - 74.6i)T - 9.40e3iT^{2} \)
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

−10.79596643783111516740533243261, −9.508307380424897777336855857416, −8.844972364071740391128651008120, −7.60683510378930639590455936084, −7.03343097877276059448062896094, −6.32125917658684063925489175893, −5.40851313333113876158819968546, −4.06902817340708033153386453717, −2.64338732202002687938990958192, −1.21037427470029303565679956703, 0.54903562392059638006835212864, 1.74970208524238918226633431363, 3.48228731916800515714852822539, 4.30490570021416163350450582633, 5.51378536866129920630457822606, 6.22052287938612580331145472066, 7.906069530450452254253319534091, 8.245704861035049793707777861633, 9.396351319459285694575164320247, 9.901290470882318567571544889720

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