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Properties

Label 2-588-7.4-c7-0-22
Degree $2$
Conductor $588$
Sign $0.266 + 0.963i$
Analytic cond. $183.682$
Root an. cond. $13.5529$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.5 + 23.3i)3-s + (−189 − 327. i)5-s + (−364.5 − 631. i)9-s + (1.24e3 − 2.15e3i)11-s − 1.48e4·13-s + 1.02e4·15-s + (−1.11e4 + 1.93e4i)17-s + (−8.15e3 − 1.41e4i)19-s + (5.75e4 + 9.97e4i)23-s + (−3.23e4 + 5.60e4i)25-s + 1.96e4·27-s + 1.57e5·29-s + (−8.22e3 + 1.42e4i)31-s + (3.35e4 + 5.80e4i)33-s + (7.46e4 + 1.29e5i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.676 − 1.17i)5-s + (−0.166 − 0.288i)9-s + (0.281 − 0.487i)11-s − 1.87·13-s + 0.780·15-s + (−0.550 + 0.953i)17-s + (−0.272 − 0.472i)19-s + (0.986 + 1.70i)23-s + (−0.414 + 0.717i)25-s + 0.192·27-s + 1.19·29-s + (−0.0496 + 0.0859i)31-s + (0.162 + 0.281i)33-s + (0.242 + 0.419i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(183.682\)
Root analytic conductor: \(13.5529\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :7/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.8487292517\)
\(L(\frac12)\) \(\approx\) \(0.8487292517\)
\(L(\frac{9}{2})\) 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 \( 1 + (13.5 - 23.3i)T \)
7 \( 1 \)
good5 \( 1 + (189 + 327. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (-1.24e3 + 2.15e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + 1.48e4T + 6.27e7T^{2} \)
17 \( 1 + (1.11e4 - 1.93e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (8.15e3 + 1.41e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-5.75e4 - 9.97e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 - 1.57e5T + 1.72e10T^{2} \)
31 \( 1 + (8.22e3 - 1.42e4i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (-7.46e4 - 1.29e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 - 2.41e5T + 1.94e11T^{2} \)
43 \( 1 + 4.43e5T + 2.71e11T^{2} \)
47 \( 1 + (-4.61e5 - 7.99e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (-3.48e5 + 6.04e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-4.35e5 + 7.53e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-1.03e6 - 1.79e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-8.40e5 + 1.45e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 1.07e6T + 9.09e12T^{2} \)
73 \( 1 + (1.20e6 - 2.08e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (1.15e6 + 1.99e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + 4.70e6T + 2.71e13T^{2} \)
89 \( 1 + (-2.07e6 - 3.58e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 - 1.62e6T + 8.07e13T^{2} \)
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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

−9.314238629204316635770853349774, −8.637010862911264115252892626707, −7.73637539807428142916597153178, −6.72867060956245636892706842375, −5.44706891027258710265698373219, −4.76121855350404147219894905692, −4.03555962712731990457970507749, −2.79435406580644740131905364675, −1.29820584673643068327865330248, −0.27287712808936817737077407743, 0.61228171173682562286892433693, 2.31671034871823264071196909022, 2.81848400884154360274088417730, 4.27780785285993913706747741657, 5.10040595528833728472585536842, 6.59028234169021416123086045370, 7.00496604969211269438897115832, 7.66906491905719259257398286289, 8.802988884851289853922672086102, 9.969891382776328576851343447943

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