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Properties

Label 2-592-148.11-c0-0-0
Degree $2$
Conductor $592$
Sign $0.957 - 0.289i$
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  + (1.5 + 0.866i)5-s + (−0.5 − 0.866i)9-s + (−1.5 + 0.866i)17-s + (1 + 1.73i)25-s − 1.73i·29-s + (0.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s − 1.73i·45-s + (−0.5 − 0.866i)49-s + (1 + 1.73i)53-s + (−1.5 − 0.866i)61-s − 2·73-s + (−0.499 + 0.866i)81-s − 3·85-s + (−1.5 + 0.866i)89-s + ⋯
L(s)  = 1  + (1.5 + 0.866i)5-s + (−0.5 − 0.866i)9-s + (−1.5 + 0.866i)17-s + (1 + 1.73i)25-s − 1.73i·29-s + (0.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s − 1.73i·45-s + (−0.5 − 0.866i)49-s + (1 + 1.73i)53-s + (−1.5 − 0.866i)61-s − 2·73-s + (−0.499 + 0.866i)81-s − 3·85-s + (−1.5 + 0.866i)89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :0),\ 0.957 - 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.075444128\)
\(L(\frac12)\) \(\approx\) \(1.075444128\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + 2T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−10.86159467130138505188479240396, −10.01861153788156159215193587461, −9.331186541139217564079014978728, −8.509610357157037397551560082876, −7.12741269597510505578483244882, −6.18203479680949645025113733492, −5.90267591917089509257768526147, −4.32606521854261524731462447939, −2.96023538580977713385986643685, −1.97004506234763376982104620581, 1.71424037287633421120349681407, 2.74636768666873062115729334995, 4.66421225838968301903200341247, 5.24412474312877977686846194968, 6.18944285318380955335176415767, 7.20997991207472386595948991662, 8.611827165421243054686561411864, 8.967622740290019631236486299381, 9.955798429273226456354226396685, 10.73596447563391279431744798671

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