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Properties

Label 2-592-37.34-c1-0-7
Degree $2$
Conductor $592$
Sign $0.998 + 0.0521i$
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  + (−2.29 + 1.92i)3-s + (1.16 + 0.423i)5-s + (−1.37 − 0.501i)7-s + (1.03 − 5.84i)9-s + (3.02 − 5.23i)11-s + (−0.652 − 3.70i)13-s + (−3.48 + 1.26i)15-s + (−0.936 + 5.30i)17-s + (2.77 − 2.32i)19-s + (4.12 − 1.50i)21-s + (2.92 + 5.07i)23-s + (−2.65 − 2.22i)25-s + (4.39 + 7.61i)27-s + (2.60 − 4.50i)29-s + 2.33·31-s + ⋯
L(s)  = 1  + (−1.32 + 1.10i)3-s + (0.520 + 0.189i)5-s + (−0.521 − 0.189i)7-s + (0.343 − 1.94i)9-s + (0.912 − 1.57i)11-s + (−0.180 − 1.02i)13-s + (−0.899 + 0.327i)15-s + (−0.227 + 1.28i)17-s + (0.635 − 0.533i)19-s + (0.899 − 0.327i)21-s + (0.610 + 1.05i)23-s + (−0.530 − 0.445i)25-s + (0.845 + 1.46i)27-s + (0.483 − 0.836i)29-s + 0.419·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.998 + 0.0521i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.998 + 0.0521i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.937750 - 0.0244601i\)
\(L(\frac12)\) \(\approx\) \(0.937750 - 0.0244601i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.43 - 2.73i)T \)
good3 \( 1 + (2.29 - 1.92i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (-1.16 - 0.423i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (1.37 + 0.501i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-3.02 + 5.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.652 + 3.70i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.936 - 5.30i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (-2.77 + 2.32i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (-2.92 - 5.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.60 + 4.50i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.33T + 31T^{2} \)
41 \( 1 + (-0.173 - 0.983i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 - 5.45T + 43T^{2} \)
47 \( 1 + (3.81 + 6.60i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.54 + 3.11i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (7.89 - 2.87i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.810 - 4.59i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (0.757 + 0.275i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-10.6 + 8.93i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 7.11T + 73T^{2} \)
79 \( 1 + (-6.61 - 2.40i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.942 + 5.34i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (5.21 - 1.89i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (5.35 + 9.28i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.68045797311217385263049085132, −9.957426636600081931997909105542, −9.308289036713100192202328238029, −8.164020246849354251016942201633, −6.56732965896653573965797679180, −6.00806931023344353080360595884, −5.35450682896020785427402767180, −4.06394832029989119243962835385, −3.20805818348863545423541603435, −0.74293730252705299474327550520, 1.21973427834495324668188788907, 2.34696036094569442975761936886, 4.45340553173839751552370635283, 5.27655715075896003541215605383, 6.42382327642804474640034119578, 6.83278053304944233501430132060, 7.61216681330711702648745984948, 9.272366426728922542824655465052, 9.677747425810671126369641901567, 10.93353690446046518461843682823

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