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Properties

Label 2-690-115.68-c1-0-23
Degree $2$
Conductor $690$
Sign $-0.795 - 0.606i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + (−1.57 − 1.59i)5-s − 1.00·6-s + (−1.37 − 1.37i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (−2.23 − 0.0144i)10-s − 0.0288i·11-s + (−0.707 + 0.707i)12-s + (0.711 + 0.711i)13-s − 1.95·14-s + (−0.0144 + 2.23i)15-s − 1.00·16-s + (−0.438 − 0.438i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (−0.702 − 0.711i)5-s − 0.408·6-s + (−0.521 − 0.521i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.707 − 0.00456i)10-s − 0.00870i·11-s + (−0.204 + 0.204i)12-s + (0.197 + 0.197i)13-s − 0.521·14-s + (−0.00372 + 0.577i)15-s − 0.250·16-s + (−0.106 − 0.106i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.795 - 0.606i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.795 - 0.606i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.196018 + 0.580194i\)
\(L(\frac12)\) \(\approx\) \(0.196018 + 0.580194i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (1.57 + 1.59i)T \)
23 \( 1 + (-3.06 - 3.68i)T \)
good7 \( 1 + (1.37 + 1.37i)T + 7iT^{2} \)
11 \( 1 + 0.0288iT - 11T^{2} \)
13 \( 1 + (-0.711 - 0.711i)T + 13iT^{2} \)
17 \( 1 + (0.438 + 0.438i)T + 17iT^{2} \)
19 \( 1 + 7.23T + 19T^{2} \)
29 \( 1 + 3.97iT - 29T^{2} \)
31 \( 1 + 5.20T + 31T^{2} \)
37 \( 1 + (-1.46 - 1.46i)T + 37iT^{2} \)
41 \( 1 + 7.17T + 41T^{2} \)
43 \( 1 + (5.83 - 5.83i)T - 43iT^{2} \)
47 \( 1 + (-6.90 + 6.90i)T - 47iT^{2} \)
53 \( 1 + (-2.73 + 2.73i)T - 53iT^{2} \)
59 \( 1 + 5.73iT - 59T^{2} \)
61 \( 1 + 5.09iT - 61T^{2} \)
67 \( 1 + (4.63 + 4.63i)T + 67iT^{2} \)
71 \( 1 + 5.47T + 71T^{2} \)
73 \( 1 + (0.908 + 0.908i)T + 73iT^{2} \)
79 \( 1 + 5.20T + 79T^{2} \)
83 \( 1 + (-9.95 + 9.95i)T - 83iT^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + (0.850 + 0.850i)T + 97iT^{2} \)
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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

−10.16496715395311943259573920400, −9.124191379166156472179292576531, −8.232178927507864274361163952137, −7.15747050139769268549187519277, −6.35191320089072630997161146538, −5.23222849045662103817072782206, −4.30879296163198325907252009427, −3.43204757513484670400890535620, −1.78843829609144164227239981308, −0.27830410178625944645891126920, 2.62473534582787048417215670852, 3.68129885647758174558305961232, 4.54889617690822862868916947891, 5.72089827765318595244724918858, 6.51902896829597751430801386015, 7.21141079809310548792046956579, 8.414953907609109630935472350476, 9.070490973494946829201369200132, 10.46984994576004375048439389560, 10.84814012198733089441954561347

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