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Properties

Label 2-693-11.4-c1-0-6
Degree $2$
Conductor $693$
Sign $-0.993 - 0.110i$
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  + (1.99 + 1.44i)2-s + (1.26 + 3.88i)4-s + (−2.80 + 2.03i)5-s + (−0.309 − 0.951i)7-s + (−1.58 + 4.89i)8-s − 8.55·10-s + (−2.91 + 1.57i)11-s + (0.528 + 0.384i)13-s + (0.762 − 2.34i)14-s + (−3.65 + 2.65i)16-s + (−0.919 + 0.668i)17-s + (−1.87 + 5.77i)19-s + (−11.4 − 8.32i)20-s + (−8.10 − 1.09i)22-s + 6.66·23-s + ⋯
L(s)  = 1  + (1.41 + 1.02i)2-s + (0.631 + 1.94i)4-s + (−1.25 + 0.911i)5-s + (−0.116 − 0.359i)7-s + (−0.561 + 1.72i)8-s − 2.70·10-s + (−0.880 + 0.474i)11-s + (0.146 + 0.106i)13-s + (0.203 − 0.626i)14-s + (−0.913 + 0.663i)16-s + (−0.223 + 0.162i)17-s + (−0.430 + 1.32i)19-s + (−2.56 − 1.86i)20-s + (−1.72 − 0.233i)22-s + 1.39·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.993 - 0.110i$
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: \((2,\ 693,\ (\ :1/2),\ -0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.119979 + 2.17002i\)
\(L(\frac12)\) \(\approx\) \(0.119979 + 2.17002i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.91 - 1.57i)T \)
good2 \( 1 + (-1.99 - 1.44i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (2.80 - 2.03i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-0.528 - 0.384i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.919 - 0.668i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.87 - 5.77i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.66T + 23T^{2} \)
29 \( 1 + (-1.41 - 4.34i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.26 + 1.64i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.135 + 0.418i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.82 + 5.61i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 + (-0.186 + 0.575i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.94 - 5.77i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.523 + 1.61i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.54 - 4.03i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.17T + 67T^{2} \)
71 \( 1 + (4.38 - 3.18i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.07 - 6.37i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.14 + 1.55i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.41 + 3.93i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 0.698T + 89T^{2} \)
97 \( 1 + (-12.0 - 8.73i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.97322453602633590430554953858, −10.40262453769198955292205392286, −8.732443189655097592092963263833, −7.58453044647223899253709286626, −7.40776487684974797565878636455, −6.48971547976662497083894798769, −5.47938127583271953514765118478, −4.38249352175883858580943537710, −3.71409200889205907212782224593, −2.77974087146392383975292270184, 0.72533052911107032105685989721, 2.53480934452181999891785586063, 3.42056539807196959003015278642, 4.55213339203988183233421395176, 4.97841851917790828685264431259, 6.04537737810462681825168430192, 7.35145262198676237215281654312, 8.440841039738486025761312208183, 9.227682530114262704464120121077, 10.57946830744356126212923265588

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