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Properties

Label 2-592-148.103-c1-0-7
Degree $2$
Conductor $592$
Sign $0.949 - 0.314i$
Analytic cond. $4.72714$
Root an. cond. $2.17419$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.536i)3-s + (−0.133 − 0.5i)5-s + (3.40 + 1.96i)7-s + (1.30 + 2.26i)9-s − 2.85·11-s + (0.473 + 1.76i)13-s + (−0.309 − 0.0829i)15-s + (5.44 + 1.45i)17-s + (−5.05 + 1.35i)19-s + (2.10 − 1.21i)21-s + (−1.66 − 1.66i)23-s + (4.09 − 2.36i)25-s + 3.47·27-s + (−0.0422 − 0.0422i)29-s + (0.427 − 0.427i)31-s + ⋯
L(s)  = 1  + (0.178 − 0.309i)3-s + (−0.0599 − 0.223i)5-s + (1.28 + 0.742i)7-s + (0.436 + 0.755i)9-s − 0.861·11-s + (0.131 + 0.489i)13-s + (−0.0799 − 0.0214i)15-s + (1.31 + 0.353i)17-s + (−1.16 + 0.311i)19-s + (0.459 − 0.265i)21-s + (−0.347 − 0.347i)23-s + (0.819 − 0.473i)25-s + 0.669·27-s + (−0.00783 − 0.00783i)29-s + (0.0767 − 0.0767i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.949 - 0.314i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.949 - 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73398 + 0.279365i\)
\(L(\frac12)\) \(\approx\) \(1.73398 + 0.279365i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.91 - 1.42i)T \)
good3 \( 1 + (-0.309 + 0.536i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.133 + 0.5i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-3.40 - 1.96i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.85T + 11T^{2} \)
13 \( 1 + (-0.473 - 1.76i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-5.44 - 1.45i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.05 - 1.35i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.66 + 1.66i)T + 23iT^{2} \)
29 \( 1 + (0.0422 + 0.0422i)T + 29iT^{2} \)
31 \( 1 + (-0.427 + 0.427i)T - 31iT^{2} \)
41 \( 1 + (-1.89 - 1.09i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.16 - 5.16i)T + 43iT^{2} \)
47 \( 1 + 13.1iT - 47T^{2} \)
53 \( 1 + (4.45 + 7.71i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.13 - 7.95i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.70 - 1.26i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-5.28 + 9.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.50 + 3.17i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.46iT - 73T^{2} \)
79 \( 1 + (15.7 - 4.22i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (6.61 - 3.82i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.76 + 10.3i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.16 - 7.16i)T - 97iT^{2} \)
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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

−10.72395551830722664872484549145, −10.00060985124687661548750504454, −8.592452426996010042637665612417, −8.193345008090938513167299138675, −7.43912079203737905490792786489, −6.08584105344991921750795511758, −5.10592412715919459317346485057, −4.33883777328764206906985279479, −2.57390996930943207306541213480, −1.59966071701882042989153083679, 1.14934848951701348918992841111, 2.85926986919703107692974375658, 4.06532865299469439005532105536, 4.91168591890594313601146566009, 6.02830141013666050004977762351, 7.37144521709221898285525103663, 7.83173905748284800124347899333, 8.867814643356846618965733832726, 9.921967936118919033290346017092, 10.66691035273771222471159985959

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