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Properties

Label 2-693-231.41-c0-0-3
Degree $2$
Conductor $693$
Sign $-0.0746 + 0.997i$
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  + (1.44 − 1.04i)2-s + (0.672 − 2.06i)4-s + (−0.951 − 0.309i)7-s + (−0.647 − 1.99i)8-s + (0.156 + 0.987i)11-s + (−1.69 + 0.550i)14-s + (−1.26 − 0.915i)16-s + (1.26 + 1.26i)22-s + 0.907i·23-s + (−0.309 − 0.951i)25-s + (−1.27 + 1.76i)28-s + (−0.610 + 1.87i)29-s − 0.680·32-s + (0.587 − 1.80i)37-s + 1.61i·43-s + (2.14 + 0.340i)44-s + ⋯
L(s)  = 1  + (1.44 − 1.04i)2-s + (0.672 − 2.06i)4-s + (−0.951 − 0.309i)7-s + (−0.647 − 1.99i)8-s + (0.156 + 0.987i)11-s + (−1.69 + 0.550i)14-s + (−1.26 − 0.915i)16-s + (1.26 + 1.26i)22-s + 0.907i·23-s + (−0.309 − 0.951i)25-s + (−1.27 + 1.76i)28-s + (−0.610 + 1.87i)29-s − 0.680·32-s + (0.587 − 1.80i)37-s + 1.61i·43-s + (2.14 + 0.340i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0746 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0746 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.0746 + 0.997i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ -0.0746 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.805884393\)
\(L(\frac12)\) \(\approx\) \(1.805884393\)
\(L(1)\) 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 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (-0.156 - 0.987i)T \)
good2 \( 1 + (-1.44 + 1.04i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 - 0.907iT - T^{2} \)
29 \( 1 + (0.610 - 1.87i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - 1.61iT - T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (1.16 + 1.59i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (0.183 - 0.253i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−10.64299459938258088785807595264, −9.884532493081300679133753244821, −9.210275745185808957845173796959, −7.53069221987970826876998788481, −6.59335445368470888233751291219, −5.71668115527645180705147482405, −4.69571426433541109716661727702, −3.83084456724513718354492078213, −2.95567458311936212579182386401, −1.71686687436873811534454076603, 2.74640021440156624103945150881, 3.60082808420528854247720005839, 4.56478611545197006528119498634, 5.80553046026705462399899002223, 6.13358018807343265620708184832, 7.08552178878075303428054059816, 8.016599392683447054595319772026, 8.919305908358385731978524818982, 9.990883115186801755959936968465, 11.25965491571078143744383333678

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