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Properties

Label 2-723-723.116-c0-0-0
Degree $2$
Conductor $723$
Sign $0.311 + 0.950i$
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  + (0.987 − 0.156i)3-s − i·4-s + (−1.47 − 1.26i)7-s + (0.951 − 0.309i)9-s + (−0.156 − 0.987i)12-s + (−0.243 + 1.01i)13-s − 16-s + (0.431 + 0.178i)19-s + (−1.65 − 1.01i)21-s + (0.951 + 0.309i)25-s + (0.891 − 0.453i)27-s + (−1.26 + 1.47i)28-s + (0.763 − 0.0600i)31-s + (−0.309 − 0.951i)36-s + (0.303 + 1.26i)37-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)3-s − i·4-s + (−1.47 − 1.26i)7-s + (0.951 − 0.309i)9-s + (−0.156 − 0.987i)12-s + (−0.243 + 1.01i)13-s − 16-s + (0.431 + 0.178i)19-s + (−1.65 − 1.01i)21-s + (0.951 + 0.309i)25-s + (0.891 − 0.453i)27-s + (−1.26 + 1.47i)28-s + (0.763 − 0.0600i)31-s + (−0.309 − 0.951i)36-s + (0.303 + 1.26i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(723\)    =    \(3 \cdot 241\)
Sign: $0.311 + 0.950i$
Analytic conductor: \(0.360824\)
Root analytic conductor: \(0.600686\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{723} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 723,\ (\ :0),\ 0.311 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.141666614\)
\(L(\frac12)\) \(\approx\) \(1.141666614\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.987 + 0.156i)T \)
241 \( 1 + (-0.987 + 0.156i)T \)
good2 \( 1 + iT^{2} \)
5 \( 1 + (-0.951 - 0.309i)T^{2} \)
7 \( 1 + (1.47 + 1.26i)T + (0.156 + 0.987i)T^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.243 - 1.01i)T + (-0.891 - 0.453i)T^{2} \)
17 \( 1 + (-0.987 + 0.156i)T^{2} \)
19 \( 1 + (-0.431 - 0.178i)T + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.453 + 0.891i)T^{2} \)
29 \( 1 + (0.587 + 0.809i)T^{2} \)
31 \( 1 + (-0.763 + 0.0600i)T + (0.987 - 0.156i)T^{2} \)
37 \( 1 + (-0.303 - 1.26i)T + (-0.891 + 0.453i)T^{2} \)
41 \( 1 + (0.587 - 0.809i)T^{2} \)
43 \( 1 + (-0.101 - 0.119i)T + (-0.156 + 0.987i)T^{2} \)
47 \( 1 + (0.587 + 0.809i)T^{2} \)
53 \( 1 + (-0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.951 + 0.309i)T^{2} \)
61 \( 1 + (0.297 + 1.87i)T + (-0.951 + 0.309i)T^{2} \)
67 \( 1 + (0.533 + 1.04i)T + (-0.587 + 0.809i)T^{2} \)
71 \( 1 + (-0.156 + 0.987i)T^{2} \)
73 \( 1 + (-0.465 - 1.93i)T + (-0.891 + 0.453i)T^{2} \)
79 \( 1 + (-0.253 + 1.59i)T + (-0.951 - 0.309i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.707 - 0.707i)T^{2} \)
97 \( 1 + (1.87 - 0.610i)T + (0.809 - 0.587i)T^{2} \)
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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

−10.12912481606747082338971319533, −9.653144204453747157730193298003, −9.036682833668613655368444930597, −7.74312178172808653011769355876, −6.73431369328193057867204133257, −6.47848789704774622811366026141, −4.80390781282762383176987738463, −3.85489533737098756267762051678, −2.79265021463431918563890455634, −1.25223685278196553726734775635, 2.65882660046340465268640877301, 2.93185861444184644926174672958, 4.01897329093518853186325579111, 5.38685374453190277382838781978, 6.57791784721061579063211544276, 7.44505103752803981703616037492, 8.376348362006575589997911500499, 9.004639742802171866572795856671, 9.648906830384746483322990006044, 10.56698923398069617682390889107

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