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Properties

Label 2-690-5.3-c2-0-42
Degree $2$
Conductor $690$
Sign $-0.582 - 0.812i$
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 + (−1.90 − 4.62i)5-s − 2.44·6-s + (2.37 − 2.37i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−6.52 − 2.72i)10-s − 8.18·11-s + (−2.44 + 2.44i)12-s + (−7.42 − 7.42i)13-s − 4.75i·14-s + (−3.33 + 7.99i)15-s − 4·16-s + (8.23 − 8.23i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (−0.380 − 0.924i)5-s − 0.408·6-s + (0.339 − 0.339i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.652 − 0.272i)10-s − 0.744·11-s + (−0.204 + 0.204i)12-s + (−0.571 − 0.571i)13-s − 0.339i·14-s + (−0.222 + 0.532i)15-s − 0.250·16-s + (0.484 − 0.484i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.582 - 0.812i$
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: \((2,\ 690,\ (\ :1),\ -0.582 - 0.812i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8086619725\)
\(L(\frac12)\) \(\approx\) \(0.8086619725\)
\(L(2)\) 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 \( 1 + (1.90 + 4.62i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good7 \( 1 + (-2.37 + 2.37i)T - 49iT^{2} \)
11 \( 1 + 8.18T + 121T^{2} \)
13 \( 1 + (7.42 + 7.42i)T + 169iT^{2} \)
17 \( 1 + (-8.23 + 8.23i)T - 289iT^{2} \)
19 \( 1 - 2.11iT - 361T^{2} \)
29 \( 1 - 15.8iT - 841T^{2} \)
31 \( 1 + 15.0T + 961T^{2} \)
37 \( 1 + (14.2 - 14.2i)T - 1.36e3iT^{2} \)
41 \( 1 - 7.01T + 1.68e3T^{2} \)
43 \( 1 + (3.12 + 3.12i)T + 1.84e3iT^{2} \)
47 \( 1 + (-29.7 + 29.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-17.6 - 17.6i)T + 2.80e3iT^{2} \)
59 \( 1 + 26.4iT - 3.48e3T^{2} \)
61 \( 1 + 28.1T + 3.72e3T^{2} \)
67 \( 1 + (66.9 - 66.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 67.4T + 5.04e3T^{2} \)
73 \( 1 + (-16.9 - 16.9i)T + 5.32e3iT^{2} \)
79 \( 1 - 75.5iT - 6.24e3T^{2} \)
83 \( 1 + (53.8 + 53.8i)T + 6.88e3iT^{2} \)
89 \( 1 + 76.6iT - 7.92e3T^{2} \)
97 \( 1 + (-34.7 + 34.7i)T - 9.40e3iT^{2} \)
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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

−9.954023402910260318241928052007, −8.831711489039108443402711878265, −7.84774401718062426089804750767, −7.17759900836135081765998358389, −5.70721241600566985240716771749, −5.12207064926585284957578917025, −4.24112452866738793946484752836, −2.89668219277037929748606752377, −1.46656558084636582640734158719, −0.25450875583570195780159502378, 2.32022138372229908607394057058, 3.49468740821201805472539606092, 4.49250335665551977698533221055, 5.46733310369788603556450571728, 6.28955668435052738540395809097, 7.29566525079922862847638708004, 7.941563838309437702754293603069, 9.073622966091141687626613885525, 10.15092329832909392156767919755, 10.84755153939543773999070016767

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