Properties

Label 2-690-23.9-c1-0-6
Degree $2$
Conductor $690$
Sign $0.686 - 0.726i$
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  + (0.959 − 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (0.841 + 0.540i)6-s + (−0.730 + 1.59i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (−0.625 − 0.183i)11-s + (0.959 + 0.281i)12-s + (0.849 + 1.86i)13-s + (−0.250 + 1.74i)14-s + (−0.654 + 0.755i)15-s + (0.415 − 0.909i)16-s + (4.93 + 3.16i)17-s + ⋯
L(s)  = 1  + (0.678 − 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (0.0636 + 0.442i)5-s + (0.343 + 0.220i)6-s + (−0.276 + 0.604i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.131 + 0.287i)10-s + (−0.188 − 0.0554i)11-s + (0.276 + 0.0813i)12-s + (0.235 + 0.516i)13-s + (−0.0668 + 0.465i)14-s + (−0.169 + 0.195i)15-s + (0.103 − 0.227i)16-s + (1.19 + 0.768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.686 - 0.726i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.686 - 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31639 + 0.998007i\)
\(L(\frac12)\) \(\approx\) \(2.31639 + 0.998007i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 + 0.281i)T \)
3 \( 1 + (-0.654 - 0.755i)T \)
5 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (-2.51 - 4.08i)T \)
good7 \( 1 + (0.730 - 1.59i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (0.625 + 0.183i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-0.849 - 1.86i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-4.93 - 3.16i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-2.65 + 1.70i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (3.61 + 2.32i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (3.53 - 4.08i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-1.47 + 10.2i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-0.146 - 1.01i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (5.55 + 6.41i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + 1.35T + 47T^{2} \)
53 \( 1 + (0.253 - 0.554i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (4.65 + 10.1i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-2.73 + 3.15i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-10.7 + 3.15i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (0.886 - 0.260i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-1.24 + 0.797i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (7.10 + 15.5i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (1.16 - 8.10i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-6.40 - 7.39i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (1.70 + 11.8i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.69503803566702528633290750337, −9.722425639885661838579935236811, −9.091232807892073746636196501455, −7.904347089539601895977530969506, −6.99891626035342257632019094639, −5.84635258741939010495182697742, −5.20703681546865384856159713439, −3.81031597949809799173767874587, −3.15587120577249847451835033793, −1.90155172068545222963713620089, 1.14927884601057329074963483647, 2.82255658180312876179381965320, 3.70871606194314842840487723083, 4.94174063383485561044029973672, 5.79181006794404411512325185973, 6.88745483144304121613840652704, 7.64160825345933049005267489683, 8.378705001631558911056603634744, 9.547442225304122113651534423144, 10.29969115871492538722020949849

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