↑

Properties

Label 2-690-15.14-c2-0-4
Degree $2$
Conductor $690$
Sign $-0.981 + 0.193i$
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  − 1.41·2-s + (−2.16 + 2.07i)3-s + 2.00·4-s + (4.21 + 2.69i)5-s + (3.06 − 2.93i)6-s + 3.87i·7-s − 2.82·8-s + (0.399 − 8.99i)9-s + (−5.95 − 3.80i)10-s − 5.04i·11-s + (−4.33 + 4.14i)12-s + 15.3i·13-s − 5.47i·14-s + (−14.7 + 2.89i)15-s + 4.00·16-s − 10.2·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.722 + 0.691i)3-s + 0.500·4-s + (0.842 + 0.538i)5-s + (0.510 − 0.488i)6-s + 0.553i·7-s − 0.353·8-s + (0.0444 − 0.999i)9-s + (−0.595 − 0.380i)10-s − 0.458i·11-s + (−0.361 + 0.345i)12-s + 1.18i·13-s − 0.391i·14-s + (−0.981 + 0.193i)15-s + 0.250·16-s − 0.602·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.981 + 0.193i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.981 + 0.193i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5195648171\)
\(L(\frac12)\) \(\approx\) \(0.5195648171\)
\(L(2)\) 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 \( 1 + (2.16 - 2.07i)T \)
5 \( 1 + (-4.21 - 2.69i)T \)
23 \( 1 - 4.79T \)
good7 \( 1 - 3.87iT - 49T^{2} \)
11 \( 1 + 5.04iT - 121T^{2} \)
13 \( 1 - 15.3iT - 169T^{2} \)
17 \( 1 + 10.2T + 289T^{2} \)
19 \( 1 + 19.1T + 361T^{2} \)
29 \( 1 - 10.8iT - 841T^{2} \)
31 \( 1 - 0.921T + 961T^{2} \)
37 \( 1 - 31.4iT - 1.36e3T^{2} \)
41 \( 1 + 28.3iT - 1.68e3T^{2} \)
43 \( 1 - 26.8iT - 1.84e3T^{2} \)
47 \( 1 + 53.0T + 2.20e3T^{2} \)
53 \( 1 + 15.1T + 2.80e3T^{2} \)
59 \( 1 + 23.8iT - 3.48e3T^{2} \)
61 \( 1 + 55.3T + 3.72e3T^{2} \)
67 \( 1 + 43.6iT - 4.48e3T^{2} \)
71 \( 1 - 2.55iT - 5.04e3T^{2} \)
73 \( 1 - 12.0iT - 5.32e3T^{2} \)
79 \( 1 + 101.T + 6.24e3T^{2} \)
83 \( 1 + 146.T + 6.88e3T^{2} \)
89 \( 1 - 126. iT - 7.92e3T^{2} \)
97 \( 1 + 136. iT - 9.40e3T^{2} \)
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

−10.73757121419315913564877301896, −9.816430446082965866517307816449, −9.180536583436860842391920989885, −8.496623660436765683020846829071, −6.88329885761589967246403948108, −6.39585960474934065570295839998, −5.53515248922731058641183608549, −4.37019296897005933918120296257, −2.95691441497707736187659827081, −1.69488978182746023968402384053, 0.25551126152523544298097012995, 1.43897812579274828273007758350, 2.52369883476892682348402961795, 4.45669306557087119131285724839, 5.52054980168674037252330636888, 6.31985976483845417222765721407, 7.17283329512882114882326527264, 8.057720206578160797865481375999, 8.886244500342217548567098446238, 10.02661669317383303757798646170

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