↑

Properties

Label 2-690-5.3-c2-0-37
Degree $2$
Conductor $690$
Sign $-0.637 + 0.770i$
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 + (3.98 − 3.01i)5-s − 2.44·6-s + (−6.00 + 6.00i)7-s + (2 + 2i)8-s + 2.99i·9-s + (−0.970 + 7.00i)10-s − 21.0·11-s + (2.44 − 2.44i)12-s + (6.18 + 6.18i)13-s − 12.0i·14-s + (8.57 + 1.18i)15-s − 4·16-s + (−1.00 + 1.00i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (0.797 − 0.603i)5-s − 0.408·6-s + (−0.857 + 0.857i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.0970 + 0.700i)10-s − 1.91·11-s + (0.204 − 0.204i)12-s + (0.476 + 0.476i)13-s − 0.857i·14-s + (0.571 + 0.0792i)15-s − 0.250·16-s + (−0.0591 + 0.0591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.637 + 0.770i$
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: \((2,\ 690,\ (\ :1),\ -0.637 + 0.770i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.009410883529\)
\(L(\frac12)\) \(\approx\) \(0.009410883529\)
\(L(2)\) 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 \( 1 + (-3.98 + 3.01i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good7 \( 1 + (6.00 - 6.00i)T - 49iT^{2} \)
11 \( 1 + 21.0T + 121T^{2} \)
13 \( 1 + (-6.18 - 6.18i)T + 169iT^{2} \)
17 \( 1 + (1.00 - 1.00i)T - 289iT^{2} \)
19 \( 1 + 17.5iT - 361T^{2} \)
29 \( 1 + 36.7iT - 841T^{2} \)
31 \( 1 + 27.5T + 961T^{2} \)
37 \( 1 + (26.3 - 26.3i)T - 1.36e3iT^{2} \)
41 \( 1 + 26.4T + 1.68e3T^{2} \)
43 \( 1 + (22.3 + 22.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (-4.99 + 4.99i)T - 2.20e3iT^{2} \)
53 \( 1 + (52.5 + 52.5i)T + 2.80e3iT^{2} \)
59 \( 1 - 59.1iT - 3.48e3T^{2} \)
61 \( 1 + 60.6T + 3.72e3T^{2} \)
67 \( 1 + (56.3 - 56.3i)T - 4.48e3iT^{2} \)
71 \( 1 + 6.19T + 5.04e3T^{2} \)
73 \( 1 + (10.2 + 10.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 44.9iT - 6.24e3T^{2} \)
83 \( 1 + (53.2 + 53.2i)T + 6.88e3iT^{2} \)
89 \( 1 + 117. iT - 7.92e3T^{2} \)
97 \( 1 + (-37.7 + 37.7i)T - 9.40e3iT^{2} \)
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

−9.867271859409491191833466305581, −8.956811383420173857570352547897, −8.551854441450366399978317905657, −7.45919012688503371371073035117, −6.27351838499812209627851458313, −5.50802875971727802743004678106, −4.73679713903834355907826849874, −3.00495204224980369726849271321, −2.04374439620295934432070056678, −0.00342645264866856456779769697, 1.65613680281808442081721359727, 2.89363797440481554718317787474, 3.49092367213298498256672287979, 5.24555759297761970945456888445, 6.32336599401749264727810977968, 7.28528377497393537246875726513, 7.898224499645923890881421112601, 8.980954040760049051788814285680, 9.919637984907876858558136412198, 10.53239921160250103664317613046

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