Properties

Label 2-693-11.4-c1-0-24
Degree $2$
Conductor $693$
Sign $0.233 + 0.972i$
Analytic cond. $5.53363$
Root an. cond. $2.35236$
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  + (0.383 + 0.278i)2-s + (−0.548 − 1.68i)4-s + (3.04 − 2.20i)5-s + (−0.309 − 0.951i)7-s + (0.552 − 1.70i)8-s + 1.78·10-s + (2.47 + 2.21i)11-s + (−2.19 − 1.59i)13-s + (0.146 − 0.450i)14-s + (−2.18 + 1.58i)16-s + (2.00 − 1.45i)17-s + (−0.539 + 1.65i)19-s + (−5.39 − 3.92i)20-s + (0.330 + 1.53i)22-s − 3.79·23-s + ⋯
L(s)  = 1  + (0.270 + 0.196i)2-s + (−0.274 − 0.844i)4-s + (1.35 − 0.988i)5-s + (−0.116 − 0.359i)7-s + (0.195 − 0.601i)8-s + 0.563·10-s + (0.744 + 0.667i)11-s + (−0.607 − 0.441i)13-s + (0.0391 − 0.120i)14-s + (−0.546 + 0.397i)16-s + (0.487 − 0.353i)17-s + (−0.123 + 0.380i)19-s + (−1.20 − 0.877i)20-s + (0.0704 + 0.327i)22-s − 0.792·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.233 + 0.972i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.233 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56186 - 1.23124i\)
\(L(\frac12)\) \(\approx\) \(1.56186 - 1.23124i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-2.47 - 2.21i)T \)
good2 \( 1 + (-0.383 - 0.278i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-3.04 + 2.20i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.19 + 1.59i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.00 + 1.45i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.539 - 1.65i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 3.79T + 23T^{2} \)
29 \( 1 + (-1.79 - 5.51i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.81 + 1.32i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.36 + 7.27i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.29 + 3.98i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.75T + 43T^{2} \)
47 \( 1 + (3.83 - 11.7i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.15 - 5.19i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.40 + 4.31i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-9.28 + 6.74i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + (-6.27 + 4.56i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.84 - 14.9i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.70 - 4.14i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-9.62 + 6.99i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.69T + 89T^{2} \)
97 \( 1 + (5.66 + 4.11i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.02306944703854832169849286441, −9.524690948532426073517106300334, −8.917289420818681639831710191629, −7.53311886927498355612218436548, −6.46709859778096211908145944294, −5.65567345092513419871915500545, −5.00805637589171128477474079783, −4.04744606630273997028714029912, −2.10594584524334194748167761159, −1.03501081414292494378680478746, 2.05152585325437507058165022212, 2.92101822260640993955084008822, 3.96041269612636011268546846195, 5.29153639795507848823928136239, 6.23479119360934536281937535372, 6.94790069051009667334842852072, 8.111865992537963559791998091793, 9.052594119718701592616995421688, 9.780850296261500455004442519920, 10.59741791308719105904063194523

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