Properties

Label 2-690-5.2-c2-0-25
Degree $2$
Conductor $690$
Sign $0.489 + 0.872i$
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 + (3.38 + 3.68i)5-s + 2.44·6-s + (−2.06 − 2.06i)7-s + (2 − 2i)8-s − 2.99i·9-s + (0.298 − 7.06i)10-s + 0.900·11-s + (−2.44 − 2.44i)12-s + (6.37 − 6.37i)13-s + 4.13i·14-s + (−8.65 − 0.365i)15-s − 4·16-s + (−18.4 − 18.4i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (0.676 + 0.736i)5-s + 0.408·6-s + (−0.295 − 0.295i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.0298 − 0.706i)10-s + 0.0818·11-s + (−0.204 − 0.204i)12-s + (0.490 − 0.490i)13-s + 0.295i·14-s + (−0.576 − 0.0243i)15-s − 0.250·16-s + (−1.08 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.872i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.489 + 0.872i$
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: \((2,\ 690,\ (\ :1),\ 0.489 + 0.872i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.111119165\)
\(L(\frac12)\) \(\approx\) \(1.111119165\)
\(L(2)\) 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 \( 1 + (-3.38 - 3.68i)T \)
23 \( 1 + (-3.39 + 3.39i)T \)
good7 \( 1 + (2.06 + 2.06i)T + 49iT^{2} \)
11 \( 1 - 0.900T + 121T^{2} \)
13 \( 1 + (-6.37 + 6.37i)T - 169iT^{2} \)
17 \( 1 + (18.4 + 18.4i)T + 289iT^{2} \)
19 \( 1 + 9.86iT - 361T^{2} \)
29 \( 1 + 7.79iT - 841T^{2} \)
31 \( 1 - 18.5T + 961T^{2} \)
37 \( 1 + (22.8 + 22.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 36.6T + 1.68e3T^{2} \)
43 \( 1 + (-14.6 + 14.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (-27.0 - 27.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (-51.0 + 51.0i)T - 2.80e3iT^{2} \)
59 \( 1 - 91.7iT - 3.48e3T^{2} \)
61 \( 1 + 44.5T + 3.72e3T^{2} \)
67 \( 1 + (21.6 + 21.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 84.1T + 5.04e3T^{2} \)
73 \( 1 + (-77.0 + 77.0i)T - 5.32e3iT^{2} \)
79 \( 1 + 31.3iT - 6.24e3T^{2} \)
83 \( 1 + (-8.70 + 8.70i)T - 6.88e3iT^{2} \)
89 \( 1 + 32.1iT - 7.92e3T^{2} \)
97 \( 1 + (104. + 104. i)T + 9.40e3iT^{2} \)
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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

−10.21714735971949510423293074583, −9.426493114473275259501916599848, −8.746246031589239441423247114356, −7.36819866329290173669603219264, −6.66867847047091900951856332369, −5.68341895253771682989826188835, −4.47192838646961855693211456403, −3.28283549912778155948268953284, −2.29313767594505219350168240965, −0.56900377216704050473536749365, 1.13128076988257180318387337637, 2.20508189228277054075521515548, 4.14043418534564591225207771048, 5.26234883340303058583327971105, 6.16681119207006020956555679241, 6.62907304162593984376393742251, 7.928210805889544230869254823799, 8.723153036155912690165682759853, 9.339935853729033337267251952940, 10.32653189342803690422525404526

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