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Properties

Label 2-588-84.11-c1-0-23
Degree $2$
Conductor $588$
Sign $0.890 - 0.455i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
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  + (0.287 − 1.38i)2-s + (1.25 + 1.19i)3-s + (−1.83 − 0.796i)4-s + (−1.08 + 0.626i)5-s + (2.01 − 1.39i)6-s + (−1.63 + 2.31i)8-s + (0.148 + 2.99i)9-s + (0.555 + 1.68i)10-s + (−2.52 + 4.38i)11-s + (−1.35 − 3.18i)12-s + 4.41·13-s + (−2.11 − 0.509i)15-s + (2.73 + 2.92i)16-s + (5.06 + 2.92i)17-s + (4.19 + 0.656i)18-s + (1.32 − 0.765i)19-s + ⋯
L(s)  = 1  + (0.203 − 0.979i)2-s + (0.724 + 0.689i)3-s + (−0.917 − 0.398i)4-s + (−0.485 + 0.280i)5-s + (0.822 − 0.569i)6-s + (−0.576 + 0.817i)8-s + (0.0494 + 0.998i)9-s + (0.175 + 0.532i)10-s + (−0.762 + 1.32i)11-s + (−0.389 − 0.920i)12-s + 1.22·13-s + (−0.544 − 0.131i)15-s + (0.682 + 0.730i)16-s + (1.22 + 0.709i)17-s + (0.987 + 0.154i)18-s + (0.304 − 0.175i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.890 - 0.455i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.890 - 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56233 + 0.376652i\)
\(L(\frac12)\) \(\approx\) \(1.56233 + 0.376652i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.287 + 1.38i)T \)
3 \( 1 + (-1.25 - 1.19i)T \)
7 \( 1 \)
good5 \( 1 + (1.08 - 0.626i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.52 - 4.38i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.41T + 13T^{2} \)
17 \( 1 + (-5.06 - 2.92i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.32 + 0.765i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.156 + 0.271i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.53iT - 29T^{2} \)
31 \( 1 + (4.24 + 2.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.66 - 4.62i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.97iT - 41T^{2} \)
43 \( 1 - 3.18iT - 43T^{2} \)
47 \( 1 + (-5.74 - 9.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.0 + 6.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.98 + 6.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.151 - 0.262i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.13 + 5.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.53T + 71T^{2} \)
73 \( 1 + (0.707 - 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6 + 3.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.78T + 83T^{2} \)
89 \( 1 + (-4.15 + 2.39i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−10.75192727843746753778870314440, −9.908124272250591899618636848442, −9.379347754814333105410348714735, −8.161744234461463486745138448265, −7.66153762154554294238976166305, −5.88239371780436108577028027519, −4.78696443250480941453117548012, −3.87970246493990340635427025977, −3.09500689917761502252400405571, −1.80657526782600554068009099074, 0.843849547116595353315506262451, 3.12481152502500309771636484281, 3.80785475313860001409298469895, 5.37488178874503707766124213169, 6.09511915535203050927366831395, 7.25695857785756841433399749948, 7.947159856090734256750407162320, 8.568328454915931144657509049863, 9.250931691449822368562377808681, 10.55435334836293517692918811950

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