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Properties

Label 2-693-77.27-c0-0-0
Degree $2$
Conductor $693$
Sign $-0.935 - 0.352i$
Analytic cond. $0.345852$
Root an. cond. $0.588091$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 1.53i)2-s + (−1.30 + 0.951i)4-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.309 + 0.951i)11-s + (−1.30 − 0.951i)14-s − 1.61·22-s + 1.61·23-s + (−0.809 − 0.587i)25-s + (0.5 − 1.53i)28-s + (0.5 − 0.363i)29-s − 0.999·32-s + (1.30 − 0.951i)37-s + 0.618·43-s + (−0.499 − 1.53i)44-s + ⋯
L(s)  = 1  + (0.5 + 1.53i)2-s + (−1.30 + 0.951i)4-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.309 + 0.951i)11-s + (−1.30 − 0.951i)14-s − 1.61·22-s + 1.61·23-s + (−0.809 − 0.587i)25-s + (0.5 − 1.53i)28-s + (0.5 − 0.363i)29-s − 0.999·32-s + (1.30 − 0.951i)37-s + 0.618·43-s + (−0.499 − 1.53i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.935 - 0.352i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ -0.935 - 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.038823066\)
\(L(\frac12)\) \(\approx\) \(1.038823066\)
\(L(1)\) 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 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
5 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 - 1.61T + T^{2} \)
29 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−11.04996261730781445822012537250, −9.861852774098504715332202052564, −9.151808767109608984813179186032, −8.207713317177649744289076363193, −7.34021348965945958675938082989, −6.62505665994720183576115830221, −5.81142417389989693553362732136, −4.96521678598967019602734835720, −3.99256638166382824567383479393, −2.54584367025632143381952511572, 1.06324680569156794495744279965, 2.75526708573957048995305573740, 3.40634035692188625996669185242, 4.42131412829208128617913367106, 5.49103255691421314764657195956, 6.62106527661525161762560285864, 7.74585662591758277278997181193, 9.025252097927546253840036284543, 9.680279837353261690387764943564, 10.59105326045868322331165086057

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