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Properties

Label 2-693-77.40-c1-0-24
Degree $2$
Conductor $693$
Sign $0.932 + 0.360i$
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  + (0.322 + 0.723i)2-s + (0.918 − 1.01i)4-s + (−1.19 + 0.126i)5-s + (2.32 − 1.26i)7-s + (2.54 + 0.825i)8-s + (−0.477 − 0.827i)10-s + (0.137 − 3.31i)11-s + (0.306 + 0.222i)13-s + (1.66 + 1.27i)14-s + (−0.0655 − 0.623i)16-s + (−0.981 − 0.437i)17-s + (−3.09 − 3.43i)19-s + (−0.972 + 1.33i)20-s + (2.44 − 0.968i)22-s + (−0.580 + 1.00i)23-s + ⋯
L(s)  = 1  + (0.227 + 0.511i)2-s + (0.459 − 0.509i)4-s + (−0.536 + 0.0563i)5-s + (0.879 − 0.476i)7-s + (0.898 + 0.291i)8-s + (−0.151 − 0.261i)10-s + (0.0414 − 0.999i)11-s + (0.0850 + 0.0618i)13-s + (0.444 + 0.341i)14-s + (−0.0163 − 0.155i)16-s + (−0.238 − 0.106i)17-s + (−0.710 − 0.788i)19-s + (−0.217 + 0.299i)20-s + (0.520 − 0.206i)22-s + (−0.121 + 0.209i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.932 + 0.360i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.932 + 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92418 - 0.359378i\)
\(L(\frac12)\) \(\approx\) \(1.92418 - 0.359378i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.32 + 1.26i)T \)
11 \( 1 + (-0.137 + 3.31i)T \)
good2 \( 1 + (-0.322 - 0.723i)T + (-1.33 + 1.48i)T^{2} \)
5 \( 1 + (1.19 - 0.126i)T + (4.89 - 1.03i)T^{2} \)
13 \( 1 + (-0.306 - 0.222i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.981 + 0.437i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (3.09 + 3.43i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (0.580 - 1.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.27 + 2.03i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.71 - 0.600i)T + (30.3 + 6.44i)T^{2} \)
37 \( 1 + (-10.3 - 2.19i)T + (33.8 + 15.0i)T^{2} \)
41 \( 1 + (-1.56 + 4.81i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 + (8.59 - 7.73i)T + (4.91 - 46.7i)T^{2} \)
53 \( 1 + (-0.981 + 9.33i)T + (-51.8 - 11.0i)T^{2} \)
59 \( 1 + (-0.851 - 0.766i)T + (6.16 + 58.6i)T^{2} \)
61 \( 1 + (0.00423 + 0.0402i)T + (-59.6 + 12.6i)T^{2} \)
67 \( 1 + (1.03 + 1.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.06 + 0.775i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.26 - 4.74i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (-0.603 - 1.35i)T + (-52.8 + 58.7i)T^{2} \)
83 \( 1 + (11.5 - 8.35i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-4.61 - 2.66i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.91 - 13.6i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.61722251698864151121060290585, −9.623246315089259411599056907376, −8.253873151090640029668232080101, −7.902357930042095001019598908139, −6.75692442900304440817757491053, −6.09083760310560757863518626882, −4.91699302220182080904768708825, −4.18580011550105864737101003438, −2.62617224093849393432123219992, −1.06425568300962965146501579057, 1.70767267576374761398113552029, 2.68889386706883259906957512390, 4.09591643240867260247564174778, 4.63623330789632076778363336251, 6.07799365080561818702243042865, 7.15187953460484760355830284172, 8.000591213691226752375963259723, 8.526148611594114238278386285081, 9.908478701682954451182720051025, 10.66032807627751284197859330050

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