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Properties

Label 2-592-37.25-c1-0-15
Degree $2$
Conductor $592$
Sign $-0.969 + 0.243i$
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  + (−1.43 − 1.20i)3-s + (−1.45 − 4.00i)5-s + (3.39 − 1.23i)7-s + (0.0923 + 0.524i)9-s + (−1.05 − 1.83i)11-s + (2.84 + 0.500i)13-s + (−2.74 + 7.53i)15-s + (0.0263 − 0.00463i)17-s + (2.07 − 2.47i)19-s + (−6.37 − 2.32i)21-s + (−2.57 − 1.48i)23-s + (−10.0 + 8.47i)25-s + (−2.31 + 4.01i)27-s + (4.96 − 2.86i)29-s + 6.76i·31-s + ⋯
L(s)  = 1  + (−0.831 − 0.697i)3-s + (−0.652 − 1.79i)5-s + (1.28 − 0.466i)7-s + (0.0307 + 0.174i)9-s + (−0.319 − 0.552i)11-s + (0.787 + 0.138i)13-s + (−0.707 + 1.94i)15-s + (0.00638 − 0.00112i)17-s + (0.476 − 0.567i)19-s + (−1.39 − 0.506i)21-s + (−0.536 − 0.309i)23-s + (−2.01 + 1.69i)25-s + (−0.446 + 0.773i)27-s + (0.921 − 0.532i)29-s + 1.21i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.969 + 0.243i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ -0.969 + 0.243i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120901 - 0.976883i\)
\(L(\frac12)\) \(\approx\) \(0.120901 - 0.976883i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.49 - 4.09i)T \)
good3 \( 1 + (1.43 + 1.20i)T + (0.520 + 2.95i)T^{2} \)
5 \( 1 + (1.45 + 4.00i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (-3.39 + 1.23i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (1.05 + 1.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.84 - 0.500i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.0263 + 0.00463i)T + (15.9 - 5.81i)T^{2} \)
19 \( 1 + (-2.07 + 2.47i)T + (-3.29 - 18.7i)T^{2} \)
23 \( 1 + (2.57 + 1.48i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.96 + 2.86i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.76iT - 31T^{2} \)
41 \( 1 + (-0.259 + 1.46i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + 5.53iT - 43T^{2} \)
47 \( 1 + (-1.30 + 2.26i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.79 + 0.652i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.92 - 5.28i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (5.65 + 0.996i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-6.50 + 2.36i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-7.10 - 5.96i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 - 16.2T + 73T^{2} \)
79 \( 1 + (-0.484 - 1.32i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.294 - 1.67i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-2.82 + 7.77i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-5.65 - 3.26i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.59685573341996221553135285693, −9.100859448826625348334329905325, −8.366425001581144402838695228758, −7.82586689220901013398183518242, −6.66394131524710797945491863153, −5.42798451087862670161907480686, −4.89668356501633464781095163708, −3.84188377715014445357737181678, −1.47580820903920806526138706627, −0.67449730909559860641612662667, 2.19383610990567782289678729312, 3.54914405669421667519391905463, 4.56887140624620092944208817216, 5.60122880106009151348546121544, 6.46774556288816781105617575969, 7.67438966587593943379529196933, 8.118021326947599740186818390821, 9.709380821830870231038324718988, 10.53253356799452986365364191356, 11.07307044879771470271038793929

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