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Properties

Label 2-723-723.662-c0-0-0
Degree $2$
Conductor $723$
Sign $0.993 + 0.114i$
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 + (0.794 − 1.29i)7-s + (0.587 − 0.809i)9-s + (0.453 + 0.891i)12-s + (−1.29 + 1.51i)13-s − 16-s + (−0.497 − 1.20i)19-s + (0.119 − 1.51i)21-s + (0.587 + 0.809i)25-s + (0.156 − 0.987i)27-s + (1.29 + 0.794i)28-s + (−1.79 + 0.431i)31-s + (0.809 + 0.587i)36-s + (1.10 + 1.29i)37-s + ⋯
L(s)  = 1  + (0.891 − 0.453i)3-s + i·4-s + (0.794 − 1.29i)7-s + (0.587 − 0.809i)9-s + (0.453 + 0.891i)12-s + (−1.29 + 1.51i)13-s − 16-s + (−0.497 − 1.20i)19-s + (0.119 − 1.51i)21-s + (0.587 + 0.809i)25-s + (0.156 − 0.987i)27-s + (1.29 + 0.794i)28-s + (−1.79 + 0.431i)31-s + (0.809 + 0.587i)36-s + (1.10 + 1.29i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(723\)    =    \(3 \cdot 241\)
Sign: $0.993 + 0.114i$
Analytic conductor: \(0.360824\)
Root analytic conductor: \(0.600686\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{723} (662, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 723,\ (\ :0),\ 0.993 + 0.114i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.280479900\)
\(L(\frac12)\) \(\approx\) \(1.280479900\)
\(L(1)\) 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 \( 1 + (-0.794 + 1.29i)T + (-0.453 - 0.891i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (1.29 - 1.51i)T + (-0.156 - 0.987i)T^{2} \)
17 \( 1 + (-0.891 + 0.453i)T^{2} \)
19 \( 1 + (0.497 + 1.20i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (0.987 - 0.156i)T^{2} \)
29 \( 1 + (-0.951 + 0.309i)T^{2} \)
31 \( 1 + (1.79 - 0.431i)T + (0.891 - 0.453i)T^{2} \)
37 \( 1 + (-1.10 - 1.29i)T + (-0.156 + 0.987i)T^{2} \)
41 \( 1 + (-0.951 - 0.309i)T^{2} \)
43 \( 1 + (-0.398 + 0.243i)T + (0.453 - 0.891i)T^{2} \)
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 \( 1 + (1.26 + 1.47i)T + (-0.156 + 0.987i)T^{2} \)
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

−10.68421810476563141283191913180, −9.388237996383681709634642665844, −8.895629371906626719897479200068, −7.82713761439501132449816539609, −7.19267660155398585408078120725, −6.84351114286082865107596557453, −4.67443911920362865457141714941, −4.16401729931998307089106151237, −2.95058599179678617891973105398, −1.78369442696156711910489067142, 1.95267202580900828802467861774, 2.73721776144824284606854038178, 4.32526139949369680532206976754, 5.32409424134504017898004553684, 5.80339518047039512211777500041, 7.40974529169361959703574485178, 8.180984389116554447044832459465, 8.987143608563911216857462901355, 9.762205806706788682811836796686, 10.44801984497171251536319242638

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