Properties

Label 2-690-5.2-c2-0-36
Degree $2$
Conductor $690$
Sign $0.331 + 0.943i$
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 + (2.69 + 4.20i)5-s − 2.44·6-s + (−5.61 − 5.61i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (−1.51 + 6.90i)10-s − 12.8·11-s + (−2.44 − 2.44i)12-s + (9.49 − 9.49i)13-s − 11.2i·14-s + (−8.46 − 1.85i)15-s − 4·16-s + (−16.0 − 16.0i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (0.539 + 0.841i)5-s − 0.408·6-s + (−0.802 − 0.802i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (−0.151 + 0.690i)10-s − 1.16·11-s + (−0.204 − 0.204i)12-s + (0.730 − 0.730i)13-s − 0.802i·14-s + (−0.564 − 0.123i)15-s − 0.250·16-s + (−0.941 − 0.941i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.331 + 0.943i$
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: \((2,\ 690,\ (\ :1),\ 0.331 + 0.943i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7063401908\)
\(L(\frac12)\) \(\approx\) \(0.7063401908\)
\(L(2)\) 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 \( 1 + (-2.69 - 4.20i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good7 \( 1 + (5.61 + 5.61i)T + 49iT^{2} \)
11 \( 1 + 12.8T + 121T^{2} \)
13 \( 1 + (-9.49 + 9.49i)T - 169iT^{2} \)
17 \( 1 + (16.0 + 16.0i)T + 289iT^{2} \)
19 \( 1 + 27.2iT - 361T^{2} \)
29 \( 1 - 10.8iT - 841T^{2} \)
31 \( 1 - 11.1T + 961T^{2} \)
37 \( 1 + (-23.3 - 23.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 48.0T + 1.68e3T^{2} \)
43 \( 1 + (-19.0 + 19.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (56.5 + 56.5i)T + 2.20e3iT^{2} \)
53 \( 1 + (40.7 - 40.7i)T - 2.80e3iT^{2} \)
59 \( 1 + 63.6iT - 3.48e3T^{2} \)
61 \( 1 - 104.T + 3.72e3T^{2} \)
67 \( 1 + (3.47 + 3.47i)T + 4.48e3iT^{2} \)
71 \( 1 + 90.8T + 5.04e3T^{2} \)
73 \( 1 + (-6.81 + 6.81i)T - 5.32e3iT^{2} \)
79 \( 1 - 103. iT - 6.24e3T^{2} \)
83 \( 1 + (-87.6 + 87.6i)T - 6.88e3iT^{2} \)
89 \( 1 - 65.1iT - 7.92e3T^{2} \)
97 \( 1 + (121. + 121. i)T + 9.40e3iT^{2} \)
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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

−10.21463499717427642858937197338, −9.420321080609970856091388606195, −8.219806728370930355436675358173, −7.00440162529236543770109251896, −6.67086524575698812903923910707, −5.57370677355724118839758235841, −4.75871666492556928547241711825, −3.45315172917835784495682788361, −2.68191208123639241133372514304, −0.21183805153944799839301856200, 1.55419847772100923160049922657, 2.51147256705792412859158444383, 3.96993028456241671452956734402, 5.06310929299879728154325718132, 6.03598763943850419922544736379, 6.32451957586404193935732862115, 7.999224910498066649720379510264, 8.777807249869462984659443848323, 9.730160926852479806807556778051, 10.45482728127975606441050256179

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