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Properties

Label 2-690-5.2-c2-0-37
Degree $2$
Conductor $690$
Sign $-0.765 - 0.642i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + (−4.46 − 2.24i)5-s + 2.44·6-s + (−8.35 − 8.35i)7-s + (2 − 2i)8-s − 2.99i·9-s + (2.21 + 6.71i)10-s + 17.4·11-s + (−2.44 − 2.44i)12-s + (11.0 − 11.0i)13-s + 16.7i·14-s + (8.22 − 2.71i)15-s − 4·16-s + (−14.9 − 14.9i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (−0.893 − 0.449i)5-s + 0.408·6-s + (−1.19 − 1.19i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.221 + 0.671i)10-s + 1.58·11-s + (−0.204 − 0.204i)12-s + (0.850 − 0.850i)13-s + 1.19i·14-s + (0.548 − 0.181i)15-s − 0.250·16-s + (−0.882 − 0.882i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.765 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.765 - 0.642i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.765 - 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2672209048\)
\(L(\frac12)\) \(\approx\) \(0.2672209048\)
\(L(2)\) 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 \( 1 + (4.46 + 2.24i)T \)
23 \( 1 + (-3.39 + 3.39i)T \)
good7 \( 1 + (8.35 + 8.35i)T + 49iT^{2} \)
11 \( 1 - 17.4T + 121T^{2} \)
13 \( 1 + (-11.0 + 11.0i)T - 169iT^{2} \)
17 \( 1 + (14.9 + 14.9i)T + 289iT^{2} \)
19 \( 1 - 2.87iT - 361T^{2} \)
29 \( 1 + 3.12iT - 841T^{2} \)
31 \( 1 - 18.1T + 961T^{2} \)
37 \( 1 + (11.8 + 11.8i)T + 1.36e3iT^{2} \)
41 \( 1 + 72.2T + 1.68e3T^{2} \)
43 \( 1 + (43.7 - 43.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (61.4 + 61.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (-43.0 + 43.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 60.4iT - 3.48e3T^{2} \)
61 \( 1 - 1.37T + 3.72e3T^{2} \)
67 \( 1 + (-60.8 - 60.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 89.3T + 5.04e3T^{2} \)
73 \( 1 + (19.0 - 19.0i)T - 5.32e3iT^{2} \)
79 \( 1 + 44.8iT - 6.24e3T^{2} \)
83 \( 1 + (1.63 - 1.63i)T - 6.88e3iT^{2} \)
89 \( 1 - 16.6iT - 7.92e3T^{2} \)
97 \( 1 + (-53.3 - 53.3i)T + 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.849946776718041970293741872021, −9.012790638959588777285189195117, −8.229084144927593176498581929330, −6.94570655653106563582804821285, −6.53049765285177718388871968607, −4.84592458832346613674794900434, −3.78083383412999414431781609684, −3.41079252736986343516664343781, −1.07659029138347281149730449056, −0.14760054614907213468921378366, 1.65678919403588979763059564712, 3.25979289474196696852731956817, 4.34722404570528481492488528019, 5.90308604713554187993287249086, 6.64750969117603944427118235242, 6.84467753951123051147959839024, 8.456929885533398784960831753096, 8.807809490204156066426750755674, 9.751922768418647224802732144533, 10.86219623625392880726427487228

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