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Properties

Label 2-723-723.434-c0-0-0
Degree $2$
Conductor $723$
Sign $0.959 - 0.281i$
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  + (0.453 + 0.891i)3-s − i·4-s + (0.0366 + 0.152i)7-s + (−0.587 + 0.809i)9-s + (0.891 − 0.453i)12-s + (1.29 − 0.101i)13-s − 16-s + (1.84 + 0.763i)19-s + (−0.119 + 0.101i)21-s + (−0.587 − 0.809i)25-s + (−0.987 − 0.156i)27-s + (0.152 − 0.0366i)28-s + (−0.652 − 0.399i)31-s + (0.809 + 0.587i)36-s + (−1.93 − 0.152i)37-s + ⋯
L(s)  = 1  + (0.453 + 0.891i)3-s − i·4-s + (0.0366 + 0.152i)7-s + (−0.587 + 0.809i)9-s + (0.891 − 0.453i)12-s + (1.29 − 0.101i)13-s − 16-s + (1.84 + 0.763i)19-s + (−0.119 + 0.101i)21-s + (−0.587 − 0.809i)25-s + (−0.987 − 0.156i)27-s + (0.152 − 0.0366i)28-s + (−0.652 − 0.399i)31-s + (0.809 + 0.587i)36-s + (−1.93 − 0.152i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(723\)    =    \(3 \cdot 241\)
Sign: $0.959 - 0.281i$
Analytic conductor: \(0.360824\)
Root analytic conductor: \(0.600686\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{723} (434, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 723,\ (\ :0),\ 0.959 - 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.118891698\)
\(L(\frac12)\) \(\approx\) \(1.118891698\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.453 - 0.891i)T \)
241 \( 1 + (-0.453 - 0.891i)T \)
good2 \( 1 + iT^{2} \)
5 \( 1 + (0.587 + 0.809i)T^{2} \)
7 \( 1 + (-0.0366 - 0.152i)T + (-0.891 + 0.453i)T^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (-1.29 + 0.101i)T + (0.987 - 0.156i)T^{2} \)
17 \( 1 + (-0.453 - 0.891i)T^{2} \)
19 \( 1 + (-1.84 - 0.763i)T + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.156 - 0.987i)T^{2} \)
29 \( 1 + (0.951 - 0.309i)T^{2} \)
31 \( 1 + (0.652 + 0.399i)T + (0.453 + 0.891i)T^{2} \)
37 \( 1 + (1.93 + 0.152i)T + (0.987 + 0.156i)T^{2} \)
41 \( 1 + (0.951 + 0.309i)T^{2} \)
43 \( 1 + (1.01 + 0.243i)T + (0.891 + 0.453i)T^{2} \)
47 \( 1 + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (-0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.587 - 0.809i)T^{2} \)
61 \( 1 + (1.04 - 0.533i)T + (0.587 - 0.809i)T^{2} \)
67 \( 1 + (0.297 - 1.87i)T + (-0.951 - 0.309i)T^{2} \)
71 \( 1 + (0.891 + 0.453i)T^{2} \)
73 \( 1 + (1.70 + 0.133i)T + (0.987 + 0.156i)T^{2} \)
79 \( 1 + (-0.550 - 0.280i)T + (0.587 + 0.809i)T^{2} \)
83 \( 1 + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (0.707 - 0.707i)T^{2} \)
97 \( 1 + (-0.533 + 0.734i)T + (-0.309 - 0.951i)T^{2} \)
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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

−10.40418463135824766417321379833, −9.934106016397133251271582852995, −9.033047625613036105843207958714, −8.363585913755258136846165142335, −7.18934607260045574477430003579, −5.83085001406882390044735737196, −5.41298846912249557847240217835, −4.18574905221693530190983243326, −3.20764836349164643119906629214, −1.68119734513961924150749710037, 1.58839849175731181443384413306, 3.11274657413009753504145779205, 3.65974515341047274721512150343, 5.21934274542044576609219742963, 6.44528910394031023683333449081, 7.28317675155151828140672102917, 7.84935860086705066778699513830, 8.826490699989378214944805889311, 9.293966590535673661570661956548, 10.78442127574942098344500837317

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