Properties

Label 2-690-23.22-c2-0-19
Degree $2$
Conductor $690$
Sign $0.885 + 0.464i$
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  + 1.41·2-s − 1.73·3-s + 2.00·4-s − 2.23i·5-s − 2.44·6-s + 3.95i·7-s + 2.82·8-s + 2.99·9-s − 3.16i·10-s − 5.67i·11-s − 3.46·12-s + 11.1·13-s + 5.59i·14-s + 3.87i·15-s + 4.00·16-s + 2.12i·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 + 0.565i·7-s + 0.353·8-s + 0.333·9-s − 0.316i·10-s − 0.515i·11-s − 0.288·12-s + 0.856·13-s + 0.399i·14-s + 0.258i·15-s + 0.250·16-s + 0.125i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.885 + 0.464i$
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: \((2,\ 690,\ (\ :1),\ 0.885 + 0.464i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.468428726\)
\(L(\frac12)\) \(\approx\) \(2.468428726\)
\(L(2)\) 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 \( 1 + (10.6 - 20.3i)T \)
good7 \( 1 - 3.95iT - 49T^{2} \)
11 \( 1 + 5.67iT - 121T^{2} \)
13 \( 1 - 11.1T + 169T^{2} \)
17 \( 1 - 2.12iT - 289T^{2} \)
19 \( 1 + 20.9iT - 361T^{2} \)
29 \( 1 - 36.0T + 841T^{2} \)
31 \( 1 - 39.7T + 961T^{2} \)
37 \( 1 + 50.2iT - 1.36e3T^{2} \)
41 \( 1 - 41.6T + 1.68e3T^{2} \)
43 \( 1 + 41.3iT - 1.84e3T^{2} \)
47 \( 1 - 60.1T + 2.20e3T^{2} \)
53 \( 1 + 3.29iT - 2.80e3T^{2} \)
59 \( 1 + 83.2T + 3.48e3T^{2} \)
61 \( 1 + 37.8iT - 3.72e3T^{2} \)
67 \( 1 - 61.1iT - 4.48e3T^{2} \)
71 \( 1 + 0.572T + 5.04e3T^{2} \)
73 \( 1 - 40.8T + 5.32e3T^{2} \)
79 \( 1 + 86.8iT - 6.24e3T^{2} \)
83 \( 1 + 84.6iT - 6.88e3T^{2} \)
89 \( 1 - 94.7iT - 7.92e3T^{2} \)
97 \( 1 - 171. iT - 9.40e3T^{2} \)
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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

−10.49728479632635219138344997884, −9.280416486681304233612806237752, −8.511891471722630989610688236242, −7.42324472963268416924022429016, −6.28606602822788575517743140340, −5.71972380560067433041783373458, −4.78453480957610241296517239159, −3.79269042844113089461728005261, −2.47713995810249520617152835703, −0.919418010424735407533268611022, 1.19381023703744341199596266885, 2.74234036859933129879207825688, 4.00272498468318482615552925007, 4.69994480752555202250773471249, 6.03128435551428707899439660523, 6.48736318989765061607192432284, 7.51447953322790244045430596347, 8.396628456638765204501748992055, 9.902682617550030818378722813738, 10.43810155284224716501126497047

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