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Properties

Label 2-592-148.99-c0-0-0
Degree $2$
Conductor $592$
Sign $0.619 - 0.785i$
Analytic cond. $0.295446$
Root an. cond. $0.543549$
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  + (0.439 + 1.20i)5-s + (0.173 + 0.984i)9-s + (−1.70 − 0.300i)13-s + (0.673 − 0.118i)17-s + (−0.500 + 0.419i)25-s + (1.70 − 0.984i)29-s + (−0.173 + 0.984i)37-s + (0.326 − 1.85i)41-s + (−1.11 + 0.642i)45-s + (0.766 − 0.642i)49-s + (−0.939 − 0.342i)53-s + (−1.93 − 0.342i)61-s + (−0.386 − 2.19i)65-s + 73-s + (−0.939 + 0.342i)81-s + ⋯
L(s)  = 1  + (0.439 + 1.20i)5-s + (0.173 + 0.984i)9-s + (−1.70 − 0.300i)13-s + (0.673 − 0.118i)17-s + (−0.500 + 0.419i)25-s + (1.70 − 0.984i)29-s + (−0.173 + 0.984i)37-s + (0.326 − 1.85i)41-s + (−1.11 + 0.642i)45-s + (0.766 − 0.642i)49-s + (−0.939 − 0.342i)53-s + (−1.93 − 0.342i)61-s + (−0.386 − 2.19i)65-s + 73-s + (−0.939 + 0.342i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.619 - 0.785i$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :0),\ 0.619 - 0.785i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9433510507\)
\(L(\frac12)\) \(\approx\) \(0.9433510507\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (0.173 - 0.984i)T \)
good3 \( 1 + (-0.173 - 0.984i)T^{2} \)
5 \( 1 + (-0.439 - 1.20i)T + (-0.766 + 0.642i)T^{2} \)
7 \( 1 + (-0.766 + 0.642i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (1.70 + 0.300i)T + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.673 + 0.118i)T + (0.939 - 0.342i)T^{2} \)
19 \( 1 + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1.70 + 0.984i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (1.93 + 0.342i)T + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.939 - 0.342i)T^{2} \)
89 \( 1 + (0.233 - 0.642i)T + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \)
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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

−10.73711922380997119287977827982, −10.23597666178039291762550623836, −9.612150068566702026200991745224, −8.173426369247757976138014363668, −7.41218601455029470261197215420, −6.66452096378253025488426385360, −5.51790675626224787473974975159, −4.59169788687953375055818823204, −2.99825697922673556365977330253, −2.23017569560954848491851908729, 1.27583242022430926378639482706, 2.89069826317162586733298811181, 4.40072641665603087822050240664, 5.10384974585126205846568748286, 6.18141969664894486687464760106, 7.21221846075302962541706325408, 8.230719323937970082359363809823, 9.290101724015431563144205198453, 9.584579275452389080614671955771, 10.62192093365466036333599531911

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