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Properties

Label 2-588-7.2-c5-0-7
Degree $2$
Conductor $588$
Sign $0.605 - 0.795i$
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 + (23.0 − 39.9i)5-s + (−40.5 + 70.1i)9-s + (315. + 546. i)11-s + 1.07e3·13-s − 415.·15-s + (80.5 + 139. i)17-s + (588. − 1.01e3i)19-s + (−1.08e3 + 1.87e3i)23-s + (499. + 864. i)25-s + 729·27-s − 4.49e3·29-s + (159. + 275. i)31-s + (2.84e3 − 4.91e3i)33-s + (−7.59e3 + 1.31e4i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.412 − 0.714i)5-s + (−0.166 + 0.288i)9-s + (0.786 + 1.36i)11-s + 1.77·13-s − 0.476·15-s + (0.0676 + 0.117i)17-s + (0.373 − 0.647i)19-s + (−0.426 + 0.738i)23-s + (0.159 + 0.276i)25-s + 0.192·27-s − 0.991·29-s + (0.0297 + 0.0515i)31-s + (0.454 − 0.786i)33-s + (−0.911 + 1.57i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(94.3056\)
Root analytic conductor: \(9.71111\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :5/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.908717483\)
\(L(\frac12)\) \(\approx\) \(1.908717483\)
\(L(\frac{7}{2})\) 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 \( 1 + (-23.0 + 39.9i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-315. - 546. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 1.07e3T + 3.71e5T^{2} \)
17 \( 1 + (-80.5 - 139. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-588. + 1.01e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.08e3 - 1.87e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 4.49e3T + 2.05e7T^{2} \)
31 \( 1 + (-159. - 275. i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (7.59e3 - 1.31e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 2.05e4T + 1.15e8T^{2} \)
43 \( 1 + 455.T + 1.47e8T^{2} \)
47 \( 1 + (1.03e4 - 1.79e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-9.65e3 - 1.67e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-3.18e3 - 5.51e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.45e4 - 4.25e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.70e4 + 2.94e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 6.29e4T + 1.80e9T^{2} \)
73 \( 1 + (4.43e3 + 7.67e3i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.72e4 - 2.98e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 7.04e3T + 3.93e9T^{2} \)
89 \( 1 + (1.01e4 - 1.75e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 5.40e4T + 8.58e9T^{2} \)
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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

−9.921022852659037381891472500301, −9.139568316011311061523957954339, −8.375508764449549254863142463599, −7.25090734595754415004215710410, −6.45157354545787009794112421254, −5.52006917310691460699254781193, −4.56825482032869761728180816621, −3.40804613057975487160854921369, −1.66493122688237954049324871011, −1.26700399564779224186915229426, 0.44926658143144563230536709623, 1.75796981814511933371357216513, 3.38470857199495160400649404749, 3.78725611736600871477000197084, 5.41196985276165885234242519960, 6.14327102457806039356258360864, 6.77168597906570616130625824040, 8.295583238786855921632599363786, 8.852365015772599403343746282036, 9.933931612422096879362135231267

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