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Properties

Label 2-592-148.119-c1-0-17
Degree $2$
Conductor $592$
Sign $-0.759 + 0.650i$
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  + (1.25 − 2.16i)3-s + (0.185 − 0.0497i)5-s + (−2.33 − 1.34i)7-s + (−1.63 − 2.83i)9-s − 5.04·11-s + (1.77 − 0.474i)13-s + (0.124 − 0.464i)15-s + (0.736 − 2.75i)17-s + (−2.04 − 7.63i)19-s + (−5.83 + 3.37i)21-s + (2.35 − 2.35i)23-s + (−4.29 + 2.48i)25-s − 0.673·27-s + (−0.248 + 0.248i)29-s + (5.30 + 5.30i)31-s + ⋯
L(s)  = 1  + (0.722 − 1.25i)3-s + (0.0830 − 0.0222i)5-s + (−0.881 − 0.508i)7-s + (−0.544 − 0.943i)9-s − 1.52·11-s + (0.491 − 0.131i)13-s + (0.0321 − 0.120i)15-s + (0.178 − 0.667i)17-s + (−0.469 − 1.75i)19-s + (−1.27 + 0.735i)21-s + (0.490 − 0.490i)23-s + (−0.859 + 0.496i)25-s − 0.129·27-s + (−0.0461 + 0.0461i)29-s + (0.952 + 0.952i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.759 + 0.650i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ -0.759 + 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.474171 - 1.28318i\)
\(L(\frac12)\) \(\approx\) \(0.474171 - 1.28318i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.30 + 2.97i)T \)
good3 \( 1 + (-1.25 + 2.16i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.185 + 0.0497i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.33 + 1.34i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 5.04T + 11T^{2} \)
13 \( 1 + (-1.77 + 0.474i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.736 + 2.75i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.04 + 7.63i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.35 + 2.35i)T - 23iT^{2} \)
29 \( 1 + (0.248 - 0.248i)T - 29iT^{2} \)
31 \( 1 + (-5.30 - 5.30i)T + 31iT^{2} \)
41 \( 1 + (-0.236 - 0.136i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.81 + 3.81i)T - 43iT^{2} \)
47 \( 1 - 5.86iT - 47T^{2} \)
53 \( 1 + (-4.57 - 7.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.71 + 0.459i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.52 - 5.70i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-5.49 + 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.13 + 5.27i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.891iT - 73T^{2} \)
79 \( 1 + (-1.92 - 7.20i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-3.22 + 1.86i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.00 - 1.34i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.21 + 7.21i)T + 97iT^{2} \)
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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

−10.36769790117217699832536858506, −9.299276660319408197387059100676, −8.475006086818415605315887443468, −7.52130068728284663441965533268, −7.01957583402688180815672988424, −6.03665108188606154662049324797, −4.73260340900363072618511582918, −3.10510453802033647608768455244, −2.44763388218276046136681324425, −0.67859226724453110230878533976, 2.39083807755092487047811646818, 3.39359555360007913966821004460, 4.23316692825718007280471063702, 5.51084566668946391202266364160, 6.25065638734524365787782305775, 7.914752416236258440140912022895, 8.394351556522715257647075312339, 9.510114934958988018676280108976, 10.05274519805953455830677913943, 10.59215056775986040609737058487

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