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Properties

Label 2-723-723.488-c0-0-0
Degree $2$
Conductor $723$
Sign $0.957 + 0.289i$
Analytic cond. $0.360824$
Root an. cond. $0.600686$
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  + (0.951 + 0.309i)3-s − 4-s + (0.278 − 1.76i)7-s + (0.809 + 0.587i)9-s + (−0.951 − 0.309i)12-s + (0.809 − 0.412i)13-s + 16-s + (−1.26 + 1.26i)19-s + (0.809 − 1.58i)21-s + (0.809 − 0.587i)25-s + (0.587 + 0.809i)27-s + (−0.278 + 1.76i)28-s + (−1.39 − 0.221i)31-s + (−0.809 − 0.587i)36-s + (0.278 + 0.142i)37-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)3-s − 4-s + (0.278 − 1.76i)7-s + (0.809 + 0.587i)9-s + (−0.951 − 0.309i)12-s + (0.809 − 0.412i)13-s + 16-s + (−1.26 + 1.26i)19-s + (0.809 − 1.58i)21-s + (0.809 − 0.587i)25-s + (0.587 + 0.809i)27-s + (−0.278 + 1.76i)28-s + (−1.39 − 0.221i)31-s + (−0.809 − 0.587i)36-s + (0.278 + 0.142i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(723\)    =    \(3 \cdot 241\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(0.360824\)
Root analytic conductor: \(0.600686\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{723} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 723,\ (\ :0),\ 0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.099587265\)
\(L(\frac12)\) \(\approx\) \(1.099587265\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.951 - 0.309i)T \)
241 \( 1 + (-0.951 - 0.309i)T \)
good2 \( 1 + T^{2} \)
5 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.278 + 1.76i)T + (-0.951 - 0.309i)T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + (-0.809 + 0.412i)T + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (-0.951 - 0.309i)T^{2} \)
19 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
23 \( 1 + (0.587 - 0.809i)T^{2} \)
29 \( 1 + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (1.39 + 0.221i)T + (0.951 + 0.309i)T^{2} \)
37 \( 1 + (-0.278 - 0.142i)T + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (-0.309 - 1.95i)T + (-0.951 + 0.309i)T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.951 - 0.309i)T^{2} \)
73 \( 1 + (1.76 + 0.896i)T + (0.587 + 0.809i)T^{2} \)
79 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
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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

−10.46283337706370619686506608383, −9.761128879310084251499752542728, −8.770538698901515850726992032913, −8.072545635261198223854167732876, −7.46738129575595143248217803066, −6.16122725979414159075499819483, −4.67670304458529449743503498533, −4.09393912646332311630487914726, −3.34588327542511403589563445496, −1.39630533678942179010839654938, 1.85374039795596091729130247879, 2.99014548955305950758618365606, 4.15474473868203239715701787957, 5.19105957145279245478180981820, 6.17836368763637249798385313567, 7.34136577894443658993419769382, 8.576738128943711369928342172824, 8.851212395661982017332590625304, 9.241240888824001516305031772649, 10.54619085970763656198857803930

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