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Properties

Label 2-592-37.26-c1-0-10
Degree $2$
Conductor $592$
Sign $0.974 + 0.223i$
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  + (−0.854 + 1.48i)3-s + (1.35 − 2.34i)5-s + (0.315 − 0.546i)7-s + (0.0391 + 0.0678i)9-s + (1 − 1.73i)13-s + (2.31 + 4.01i)15-s + (−2.13 − 3.69i)17-s + (2.63 − 4.55i)19-s + (0.539 + 0.933i)21-s + 4.04·23-s + (−1.17 − 2.02i)25-s − 5.26·27-s + 7.04·29-s + 0.581·31-s + (−0.854 − 1.48i)35-s + ⋯
L(s)  = 1  + (−0.493 + 0.854i)3-s + (0.605 − 1.04i)5-s + (0.119 − 0.206i)7-s + (0.0130 + 0.0226i)9-s + (0.277 − 0.480i)13-s + (0.597 + 1.03i)15-s + (−0.516 − 0.895i)17-s + (0.603 − 1.04i)19-s + (0.117 + 0.203i)21-s + 0.844·23-s + (−0.234 − 0.405i)25-s − 1.01·27-s + 1.30·29-s + 0.104·31-s + (−0.144 − 0.250i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.974 + 0.223i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.974 + 0.223i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42147 - 0.161204i\)
\(L(\frac12)\) \(\approx\) \(1.42147 - 0.161204i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.06 - 0.478i)T \)
good3 \( 1 + (0.854 - 1.48i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.35 + 2.34i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.315 + 0.546i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.13 + 3.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.63 + 4.55i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.04T + 23T^{2} \)
29 \( 1 - 7.04T + 29T^{2} \)
31 \( 1 - 0.581T + 31T^{2} \)
41 \( 1 + (2.67 - 4.62i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 7.86T + 43T^{2} \)
47 \( 1 + 3.41T + 47T^{2} \)
53 \( 1 + (-0.460 - 0.798i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.63 + 4.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.47 - 2.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.41 + 9.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.65 - 8.06i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.23T + 73T^{2} \)
79 \( 1 + (-1.55 + 2.68i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.64 + 9.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.71 - 9.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−10.70479846054130527292360497270, −9.629006809500337503893518153193, −9.226560828790540390966447185524, −8.170186516126977518191322435392, −7.01737558000210642767626094362, −5.78307491649110854671503793810, −4.91845401761191050614843456374, −4.48660207323322217742001825511, −2.82062775879862827604171924313, −1.00703080508782964588271315256, 1.42288740031147540938491375336, 2.63381644048816298858926381931, 3.99763374339318140238026130245, 5.55991396854306523187091930677, 6.36676409723129542631067188308, 6.86220609049534051412303892755, 7.87367920354795436842248973506, 8.978638761995177944038371781442, 10.02563546217884150504977719649, 10.74856974624749030436862305822

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