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Properties

Label 2-723-723.644-c0-0-0
Degree $2$
Conductor $723$
Sign $-0.0817 + 0.996i$
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  + (−0.891 − 0.453i)3-s − i·4-s + (1.10 − 0.678i)7-s + (0.587 + 0.809i)9-s + (−0.453 + 0.891i)12-s + (0.119 − 0.101i)13-s − 16-s + (−1.40 − 0.581i)19-s + (−1.29 + 0.101i)21-s + (0.587 − 0.809i)25-s + (−0.156 − 0.987i)27-s + (−0.678 − 1.10i)28-s + (0.178 − 0.744i)31-s + (0.809 − 0.587i)36-s + (0.794 + 0.678i)37-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)3-s − i·4-s + (1.10 − 0.678i)7-s + (0.587 + 0.809i)9-s + (−0.453 + 0.891i)12-s + (0.119 − 0.101i)13-s − 16-s + (−1.40 − 0.581i)19-s + (−1.29 + 0.101i)21-s + (0.587 − 0.809i)25-s + (−0.156 − 0.987i)27-s + (−0.678 − 1.10i)28-s + (0.178 − 0.744i)31-s + (0.809 − 0.587i)36-s + (0.794 + 0.678i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0817 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0817 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(723\)    =    \(3 \cdot 241\)
Sign: $-0.0817 + 0.996i$
Analytic conductor: \(0.360824\)
Root analytic conductor: \(0.600686\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{723} (644, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 723,\ (\ :0),\ -0.0817 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7716929238\)
\(L(\frac12)\) \(\approx\) \(0.7716929238\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.891 + 0.453i)T \)
241 \( 1 + (0.891 + 0.453i)T \)
good2 \( 1 + iT^{2} \)
5 \( 1 + (-0.587 + 0.809i)T^{2} \)
7 \( 1 + (-1.10 + 0.678i)T + (0.453 - 0.891i)T^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.119 + 0.101i)T + (0.156 - 0.987i)T^{2} \)
17 \( 1 + (0.891 + 0.453i)T^{2} \)
19 \( 1 + (1.40 + 0.581i)T + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.987 - 0.156i)T^{2} \)
29 \( 1 + (-0.951 - 0.309i)T^{2} \)
31 \( 1 + (-0.178 + 0.744i)T + (-0.891 - 0.453i)T^{2} \)
37 \( 1 + (-0.794 - 0.678i)T + (0.156 + 0.987i)T^{2} \)
41 \( 1 + (-0.951 + 0.309i)T^{2} \)
43 \( 1 + (1.01 - 1.65i)T + (-0.453 - 0.891i)T^{2} \)
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 \( 1 + (0.533 - 1.04i)T + (-0.587 - 0.809i)T^{2} \)
67 \( 1 + (-1.87 + 0.297i)T + (0.951 - 0.309i)T^{2} \)
71 \( 1 + (-0.453 - 0.891i)T^{2} \)
73 \( 1 + (0.355 + 0.303i)T + (0.156 + 0.987i)T^{2} \)
79 \( 1 + (0.280 + 0.550i)T + (-0.587 + 0.809i)T^{2} \)
83 \( 1 + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.707 - 0.707i)T^{2} \)
97 \( 1 + (-1.04 - 1.44i)T + (-0.309 + 0.951i)T^{2} \)
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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

−10.59236604923278222734838679230, −9.856178569229682505693897391662, −8.556009321180918711157321412218, −7.69703502059571623871248520025, −6.64278376436962419704114948983, −6.06464926050435510047230666982, −4.83672008349479713335453484206, −4.47276930001084073944882899195, −2.18529203568086938100292936089, −1.01271942470610658630927416191, 1.99465812269000676796825668164, 3.55419314435999164149829661703, 4.52408111059706711549852476155, 5.31935475000001675500800220688, 6.40666428271411134011395790422, 7.34840319677435545766726690842, 8.427918146669724283954347342337, 8.909760272490771673139785701845, 10.13522008364931495249269625787, 11.11951647867829258631039189951

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