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Properties

Label 2-690-5.2-c2-0-4
Degree $2$
Conductor $690$
Sign $-0.641 - 0.766i$
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 + (4.00 + 2.99i)5-s + 2.44·6-s + (−4.65 − 4.65i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−1.01 − 6.99i)10-s − 0.959·11-s + (−2.44 − 2.44i)12-s + (−7.15 + 7.15i)13-s + 9.31i·14-s + (−8.57 + 1.23i)15-s − 4·16-s + (17.8 + 17.8i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (0.800 + 0.598i)5-s + 0.408·6-s + (−0.665 − 0.665i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.101 − 0.699i)10-s − 0.0871·11-s + (−0.204 − 0.204i)12-s + (−0.550 + 0.550i)13-s + 0.665i·14-s + (−0.571 + 0.0824i)15-s − 0.250·16-s + (1.04 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.641 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.641 - 0.766i$
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: \((2,\ 690,\ (\ :1),\ -0.641 - 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5564076985\)
\(L(\frac12)\) \(\approx\) \(0.5564076985\)
\(L(2)\) 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 \( 1 + (-4.00 - 2.99i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good7 \( 1 + (4.65 + 4.65i)T + 49iT^{2} \)
11 \( 1 + 0.959T + 121T^{2} \)
13 \( 1 + (7.15 - 7.15i)T - 169iT^{2} \)
17 \( 1 + (-17.8 - 17.8i)T + 289iT^{2} \)
19 \( 1 + 11.9iT - 361T^{2} \)
29 \( 1 + 10.1iT - 841T^{2} \)
31 \( 1 + 17.8T + 961T^{2} \)
37 \( 1 + (-35.6 - 35.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 70.4T + 1.68e3T^{2} \)
43 \( 1 + (32.3 - 32.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (36.1 + 36.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (22.0 - 22.0i)T - 2.80e3iT^{2} \)
59 \( 1 - 7.49iT - 3.48e3T^{2} \)
61 \( 1 + 83.1T + 3.72e3T^{2} \)
67 \( 1 + (-36.1 - 36.1i)T + 4.48e3iT^{2} \)
71 \( 1 + 19.9T + 5.04e3T^{2} \)
73 \( 1 + (39.5 - 39.5i)T - 5.32e3iT^{2} \)
79 \( 1 - 26.1iT - 6.24e3T^{2} \)
83 \( 1 + (38.2 - 38.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 67.4iT - 7.92e3T^{2} \)
97 \( 1 + (-90.5 - 90.5i)T + 9.40e3iT^{2} \)
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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

−10.27788163592348037953779441217, −9.987929567110139562170511509883, −9.263534686149134744191113149800, −8.041245699955885253634934053783, −6.94803256183590526268136095871, −6.30346776259771819259700447586, −5.12330399906604113724478611697, −3.84854716455146511583438820859, −2.91671048806906425469962532477, −1.49789266971982689063345987697, 0.24653947803876967817444882466, 1.67970997590927053748423400124, 2.99645308361844422916672711619, 4.94141179259148005474569302417, 5.59941905065617532777294133810, 6.29260925605369352080829932315, 7.33473719967614953041415401717, 8.187123510566281736323463645139, 9.198817284908891871002862842998, 9.784339112907125804727939687131

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