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Properties

Label 2-693-231.41-c0-0-1
Degree $2$
Conductor $693$
Sign $0.915 + 0.402i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.915 + 0.402i$
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: \((2,\ 693,\ (\ :0),\ 0.915 + 0.402i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3547899557\)
\(L(\frac12)\) \(\approx\) \(0.3547899557\)
\(L(1)\) 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

−10.31458672137530169813646691655, −9.593077544384556870978776908902, −8.826752949766125449138360006151, −8.041314114754587497754515148825, −7.29459194061942389160496022399, −6.14170304747976324656167035471, −6.00711249706515328426987156202, −4.26312641794211840402204527469, −2.63805802925140126106760694773, −0.63545503605285065932662652634, 1.57709066657913199015786031238, 2.79345917471770236625237965966, 3.68466531739455021112899871360, 5.27749927515240420867514017294, 6.78620172872381486498921003079, 7.40846554457306675037913149212, 8.512666702152242306642063934290, 9.197672470766826269496999451744, 9.932704549401755871653071963840, 10.39792223013185484570675186820

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