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Properties

Label 2-693-11.9-c1-0-2
Degree $2$
Conductor $693$
Sign $-0.998 + 0.0547i$
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  + (−0.788 + 2.42i)2-s + (−3.65 − 2.65i)4-s + (−1.05 − 3.26i)5-s + (−0.809 − 0.587i)7-s + (5.18 − 3.76i)8-s + 8.75·10-s + (0.0297 + 3.31i)11-s + (−0.672 + 2.06i)13-s + (2.06 − 1.49i)14-s + (2.26 + 6.97i)16-s + (1.39 + 4.29i)17-s + (1.96 − 1.42i)19-s + (−4.78 + 14.7i)20-s + (−8.07 − 2.54i)22-s − 0.648·23-s + ⋯
L(s)  = 1  + (−0.557 + 1.71i)2-s + (−1.82 − 1.32i)4-s + (−0.473 − 1.45i)5-s + (−0.305 − 0.222i)7-s + (1.83 − 1.33i)8-s + 2.76·10-s + (0.00897 + 0.999i)11-s + (−0.186 + 0.573i)13-s + (0.551 − 0.400i)14-s + (0.566 + 1.74i)16-s + (0.338 + 1.04i)17-s + (0.449 − 0.326i)19-s + (−1.06 + 3.29i)20-s + (−1.72 − 0.542i)22-s − 0.135·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.998 + 0.0547i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ -0.998 + 0.0547i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0137264 - 0.501428i\)
\(L(\frac12)\) \(\approx\) \(0.0137264 - 0.501428i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-0.0297 - 3.31i)T \)
good2 \( 1 + (0.788 - 2.42i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (1.05 + 3.26i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.672 - 2.06i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.39 - 4.29i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.96 + 1.42i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.648T + 23T^{2} \)
29 \( 1 + (1.01 + 0.736i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.48 - 7.64i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.05 + 2.94i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.12 - 1.54i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.46T + 43T^{2} \)
47 \( 1 + (4.07 - 2.96i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.14 - 12.7i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.20 - 4.50i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.42 - 13.6i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 6.22T + 67T^{2} \)
71 \( 1 + (1.30 + 4.02i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.79 - 3.48i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.01 + 9.28i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.58 - 7.96i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 4.76T + 89T^{2} \)
97 \( 1 + (2.69 - 8.27i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.45495984527906991852970602749, −9.540280145070476137415146112347, −8.966774363691816317812534671568, −8.236808265126576813871414418676, −7.44809283493898046039712163375, −6.71464386201125800850510098129, −5.57425899460305205760152814395, −4.82105017990052784531075318613, −4.03085120850645817771013816730, −1.31769905208989281753633268975, 0.35271257959197038152938814155, 2.26063302256924063709212464886, 3.26836547126343656774603305365, 3.58728254092950280057231286125, 5.27051791458754866025515690935, 6.63597178005910678984213495546, 7.73068704835285626814132805184, 8.461055318024548891959769394585, 9.688306898640666899817434745889, 10.03304470013484766995067199890

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