Properties

Label 2-690-23.22-c2-0-15
Degree $2$
Conductor $690$
Sign $0.694 - 0.719i$
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 about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 1.73·3-s + 2.00·4-s + 2.23i·5-s + 2.44·6-s + 6.64i·7-s + 2.82·8-s + 2.99·9-s + 3.16i·10-s − 9.54i·11-s + 3.46·12-s + 10.3·13-s + 9.39i·14-s + 3.87i·15-s + 4.00·16-s + 15.5i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.500·4-s + 0.447i·5-s + 0.408·6-s + 0.949i·7-s + 0.353·8-s + 0.333·9-s + 0.316i·10-s − 0.867i·11-s + 0.288·12-s + 0.794·13-s + 0.671i·14-s + 0.258i·15-s + 0.250·16-s + 0.917i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.694 - 0.719i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.694 - 0.719i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.642765897\)
\(L(\frac12)\) \(\approx\) \(3.642765897\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 - 1.73T \)
5 \( 1 - 2.23iT \)
23 \( 1 + (-16.5 - 15.9i)T \)
good7 \( 1 - 6.64iT - 49T^{2} \)
11 \( 1 + 9.54iT - 121T^{2} \)
13 \( 1 - 10.3T + 169T^{2} \)
17 \( 1 - 15.5iT - 289T^{2} \)
19 \( 1 + 0.664iT - 361T^{2} \)
29 \( 1 - 48.3T + 841T^{2} \)
31 \( 1 + 42.5T + 961T^{2} \)
37 \( 1 - 37.9iT - 1.36e3T^{2} \)
41 \( 1 + 32.8T + 1.68e3T^{2} \)
43 \( 1 - 56.3iT - 1.84e3T^{2} \)
47 \( 1 - 26.7T + 2.20e3T^{2} \)
53 \( 1 - 13.8iT - 2.80e3T^{2} \)
59 \( 1 + 92.5T + 3.48e3T^{2} \)
61 \( 1 + 114. iT - 3.72e3T^{2} \)
67 \( 1 + 119. iT - 4.48e3T^{2} \)
71 \( 1 - 104.T + 5.04e3T^{2} \)
73 \( 1 - 4.46T + 5.32e3T^{2} \)
79 \( 1 + 29.4iT - 6.24e3T^{2} \)
83 \( 1 + 136. iT - 6.88e3T^{2} \)
89 \( 1 + 19.1iT - 7.92e3T^{2} \)
97 \( 1 - 3.15iT - 9.40e3T^{2} \)
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

−10.61586707755425100485994871854, −9.410763407258325481044405056665, −8.567630027617534967124707995468, −7.86540621222469855035315955723, −6.56652652107144260478283723060, −5.97947701426937597994421713321, −4.89766492828397085591556554330, −3.55027011463768739333348226316, −2.95200298603393475261479610740, −1.62355261377803933524575918272, 1.06540456857482490400090152796, 2.47115131240259722848726034833, 3.73519106920953924781601856279, 4.46892646590057546996474534573, 5.42307905227906671473508250932, 6.83016450213382637026597793520, 7.29453763549096508360547530360, 8.409395092716184378786541522168, 9.260775163793954671685220521699, 10.28493370537970382043622882428

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