Properties

Label 2-592-148.67-c0-0-0
Degree $2$
Conductor $592$
Sign $0.942 - 0.334i$
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  + (−0.673 + 0.118i)5-s + (0.766 + 0.642i)9-s + (1.11 + 1.32i)13-s + (1.26 − 1.50i)17-s + (−0.500 + 0.181i)25-s + (−1.11 − 0.642i)29-s + (−0.766 + 0.642i)37-s + (−0.266 + 0.223i)41-s + (−0.592 − 0.342i)45-s + (−0.939 + 0.342i)49-s + (0.173 − 0.984i)53-s + (−0.826 − 0.984i)61-s + (−0.907 − 0.761i)65-s + 73-s + (0.173 + 0.984i)81-s + ⋯
L(s)  = 1  + (−0.673 + 0.118i)5-s + (0.766 + 0.642i)9-s + (1.11 + 1.32i)13-s + (1.26 − 1.50i)17-s + (−0.500 + 0.181i)25-s + (−1.11 − 0.642i)29-s + (−0.766 + 0.642i)37-s + (−0.266 + 0.223i)41-s + (−0.592 − 0.342i)45-s + (−0.939 + 0.342i)49-s + (0.173 − 0.984i)53-s + (−0.826 − 0.984i)61-s + (−0.907 − 0.761i)65-s + 73-s + (0.173 + 0.984i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.942 - 0.334i$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :0),\ 0.942 - 0.334i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8950703145\)
\(L(\frac12)\) \(\approx\) \(0.8950703145\)
\(L(1)\) 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 \( 1 + (0.766 - 0.642i)T \)
good3 \( 1 + (-0.766 - 0.642i)T^{2} \)
5 \( 1 + (0.673 - 0.118i)T + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.939 - 0.342i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.11 - 1.32i)T + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \)
19 \( 1 + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.11 + 0.642i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (0.826 + 0.984i)T + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.173 - 0.984i)T^{2} \)
89 \( 1 + (1.93 + 0.342i)T + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2} \)
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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

−11.19999509415452138682016670002, −9.979762501229987188374107141856, −9.317291420422317572628053084852, −8.149954891650736220485821897350, −7.44107899532965075407921531019, −6.61419294587995067397108551881, −5.33063779785418827055093916835, −4.29854597402642280072217352899, −3.36209230845562546230434852951, −1.69428104488124647492310375614, 1.36316111834038592372152018835, 3.45118103062819839022177575889, 3.91980819586604418465150185011, 5.44367354334518441885365385300, 6.24885655882283037685613687615, 7.49252975332616077752233453450, 8.110829825287575311052649244827, 9.016933298363339119305496902966, 10.17867218968191143833308191308, 10.69011055671439880650225803600

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