Properties

Label 2-588-7.4-c5-0-4
Degree $2$
Conductor $588$
Sign $-0.991 + 0.126i$
Analytic cond. $94.3056$
Root an. cond. $9.71111$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 + 7.79i)3-s + (−3.19 − 5.53i)5-s + (−40.5 − 70.1i)9-s + (−372. + 645. i)11-s + 879.·13-s + 57.5·15-s + (−987. + 1.71e3i)17-s + (484. + 838. i)19-s + (−903. − 1.56e3i)23-s + (1.54e3 − 2.67e3i)25-s + 729·27-s + 547.·29-s + (3.16e3 − 5.48e3i)31-s + (−3.35e3 − 5.81e3i)33-s + (7.31e3 + 1.26e4i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.0571 − 0.0990i)5-s + (−0.166 − 0.288i)9-s + (−0.928 + 1.60i)11-s + 1.44·13-s + 0.0660·15-s + (−0.828 + 1.43i)17-s + (0.307 + 0.533i)19-s + (−0.356 − 0.616i)23-s + (0.493 − 0.854i)25-s + 0.192·27-s + 0.120·29-s + (0.591 − 1.02i)31-s + (−0.536 − 0.928i)33-s + (0.878 + 1.52i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(94.3056\)
Root analytic conductor: \(9.71111\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :5/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8309474812\)
\(L(\frac12)\) \(\approx\) \(0.8309474812\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 \)
good5 \( 1 + (3.19 + 5.53i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (372. - 645. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 879.T + 3.71e5T^{2} \)
17 \( 1 + (987. - 1.71e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-484. - 838. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (903. + 1.56e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 547.T + 2.05e7T^{2} \)
31 \( 1 + (-3.16e3 + 5.48e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-7.31e3 - 1.26e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.49e4T + 1.15e8T^{2} \)
43 \( 1 - 1.66e4T + 1.47e8T^{2} \)
47 \( 1 + (-3.82e3 - 6.63e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.19e4 - 2.07e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (7.69e3 - 1.33e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.36e4 + 2.36e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-259. + 449. i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 1.00e4T + 1.80e9T^{2} \)
73 \( 1 + (3.28e4 - 5.68e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.96e4 - 3.41e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 3.55e4T + 3.93e9T^{2} \)
89 \( 1 + (5.43e4 + 9.41e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.41e5T + 8.58e9T^{2} \)
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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

−10.39409842133307349096830975708, −9.695164859632438654329756136654, −8.526710198159198833983630479142, −7.918124135546246030417813700125, −6.58349785322110624009326115302, −5.90048281993121371540585012880, −4.60874584822970672990223963841, −4.08043886184070123937452972345, −2.61467855822044244080536802031, −1.38199314076074619207557439609, 0.21088160449774527037640047312, 1.11791852343256181328585960884, 2.65027710115770587717027144774, 3.52678588711694396282924451067, 5.03829758285118149698768675051, 5.81274295094769446396065972992, 6.71897629492152640817660761687, 7.66617921570786843706170446294, 8.573301741540568672544748944522, 9.253687809098283748386124346930

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