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Properties

Label 2-592-148.103-c1-0-1
Degree $2$
Conductor $592$
Sign $-0.474 - 0.880i$
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  + (−0.436 + 0.756i)3-s + (−0.192 − 0.717i)5-s + (−1.90 − 1.09i)7-s + (1.11 + 1.93i)9-s − 1.68·11-s + (1.56 + 5.83i)13-s + (0.626 + 0.167i)15-s + (−4.60 − 1.23i)17-s + (−3.93 + 1.05i)19-s + (1.66 − 0.959i)21-s + (5.76 + 5.76i)23-s + (3.85 − 2.22i)25-s − 4.57·27-s + (5.98 + 5.98i)29-s + (−7.55 + 7.55i)31-s + ⋯
L(s)  = 1  + (−0.252 + 0.436i)3-s + (−0.0859 − 0.320i)5-s + (−0.719 − 0.415i)7-s + (0.372 + 0.645i)9-s − 0.509·11-s + (0.433 + 1.61i)13-s + (0.161 + 0.0433i)15-s + (−1.11 − 0.299i)17-s + (−0.901 + 0.241i)19-s + (0.362 − 0.209i)21-s + (1.20 + 1.20i)23-s + (0.770 − 0.444i)25-s − 0.880·27-s + (1.11 + 1.11i)29-s + (−1.35 + 1.35i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.474 - 0.880i$
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: \((2,\ 592,\ (\ :1/2),\ -0.474 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.421599 + 0.706024i\)
\(L(\frac12)\) \(\approx\) \(0.421599 + 0.706024i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.44 - 2.71i)T \)
good3 \( 1 + (0.436 - 0.756i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.192 + 0.717i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.90 + 1.09i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.68T + 11T^{2} \)
13 \( 1 + (-1.56 - 5.83i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.60 + 1.23i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.93 - 1.05i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-5.76 - 5.76i)T + 23iT^{2} \)
29 \( 1 + (-5.98 - 5.98i)T + 29iT^{2} \)
31 \( 1 + (7.55 - 7.55i)T - 31iT^{2} \)
41 \( 1 + (4.64 + 2.68i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.512 - 0.512i)T + 43iT^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 + (-0.799 - 1.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.25 - 12.1i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.62 - 0.704i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.515 + 0.892i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.02 - 1.16i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.19iT - 73T^{2} \)
79 \( 1 + (-11.6 + 3.12i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-5.53 + 3.19i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.03 + 3.86i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-10.1 + 10.1i)T - 97iT^{2} \)
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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

−10.73850110199416324951818442437, −10.34646294792033638556544685514, −9.077353115548995751315917257458, −8.679899544339601641582672470371, −7.08113844075843160126137899192, −6.73873195277391718045810828085, −5.20487336802582780175916305458, −4.52142082087285185034265900314, −3.42986381857236024231678963426, −1.79072165281557171331603572590, 0.46367285662922589745637206582, 2.46113057987152773943548831494, 3.48816498259736682025350606954, 4.85994647977430718035391872651, 6.15900676866587884649860240316, 6.55932381400784668772620614309, 7.67777471419378393928619195576, 8.648207477088217042264232097720, 9.498365902768802991457996402642, 10.63870556488658239958611022459

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