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Properties

Label 2-723-723.47-c0-0-0
Degree $2$
Conductor $723$
Sign $0.697 - 0.716i$
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  + (−0.156 − 0.987i)3-s + i·4-s + (0.133 + 1.70i)7-s + (−0.951 + 0.309i)9-s + (0.987 − 0.156i)12-s + (0.243 + 0.398i)13-s − 16-s + (0.399 + 0.965i)19-s + (1.65 − 0.398i)21-s + (−0.951 − 0.309i)25-s + (0.453 + 0.891i)27-s + (−1.70 + 0.133i)28-s + (1.20 − 1.40i)31-s + (−0.309 − 0.951i)36-s + (1.04 − 1.70i)37-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)3-s + i·4-s + (0.133 + 1.70i)7-s + (−0.951 + 0.309i)9-s + (0.987 − 0.156i)12-s + (0.243 + 0.398i)13-s − 16-s + (0.399 + 0.965i)19-s + (1.65 − 0.398i)21-s + (−0.951 − 0.309i)25-s + (0.453 + 0.891i)27-s + (−1.70 + 0.133i)28-s + (1.20 − 1.40i)31-s + (−0.309 − 0.951i)36-s + (1.04 − 1.70i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(723\)    =    \(3 \cdot 241\)
Sign: $0.697 - 0.716i$
Analytic conductor: \(0.360824\)
Root analytic conductor: \(0.600686\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{723} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 723,\ (\ :0),\ 0.697 - 0.716i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8652508337\)
\(L(\frac12)\) \(\approx\) \(0.8652508337\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.156 + 0.987i)T \)
241 \( 1 + (0.156 + 0.987i)T \)
good2 \( 1 - iT^{2} \)
5 \( 1 + (0.951 + 0.309i)T^{2} \)
7 \( 1 + (-0.133 - 1.70i)T + (-0.987 + 0.156i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.243 - 0.398i)T + (-0.453 + 0.891i)T^{2} \)
17 \( 1 + (0.156 + 0.987i)T^{2} \)
19 \( 1 + (-0.399 - 0.965i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.891 - 0.453i)T^{2} \)
29 \( 1 + (-0.587 - 0.809i)T^{2} \)
31 \( 1 + (-1.20 + 1.40i)T + (-0.156 - 0.987i)T^{2} \)
37 \( 1 + (-1.04 + 1.70i)T + (-0.453 - 0.891i)T^{2} \)
41 \( 1 + (-0.587 + 0.809i)T^{2} \)
43 \( 1 + (-1.51 - 0.119i)T + (0.987 + 0.156i)T^{2} \)
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 \( 1 + (1.87 - 0.297i)T + (0.951 - 0.309i)T^{2} \)
67 \( 1 + (-1.04 + 0.533i)T + (0.587 - 0.809i)T^{2} \)
71 \( 1 + (0.987 + 0.156i)T^{2} \)
73 \( 1 + (0.678 - 1.10i)T + (-0.453 - 0.891i)T^{2} \)
79 \( 1 + (1.59 + 0.253i)T + (0.951 + 0.309i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (-0.707 - 0.707i)T^{2} \)
97 \( 1 + (0.297 - 0.0966i)T + (0.809 - 0.587i)T^{2} \)
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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

−11.19456804020167040888595813385, −9.566683469448779752462465487995, −8.806491984731707971353226407300, −8.048065311518702400557208015365, −7.47644321001915764659500458176, −6.15834257755897216542431764634, −5.72970259438718359845577324556, −4.22882190836424404099183446444, −2.81656968986565822336893575832, −2.05976409916572979869435350999, 1.02622309788362168630356277211, 3.06906299133848533871849415694, 4.30491891256682689586403455735, 4.86248186453636296378591173364, 5.97557798800104335784291095761, 6.86732885777371112634794612677, 7.936657869138475327964445399940, 9.090880361220743126809702509995, 9.920053637902690417561433928844, 10.41789119464286245734899672643

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