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Properties

Label 2-588-7.2-c5-0-12
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 + (−46.4 + 80.3i)5-s + (−40.5 + 70.1i)9-s + (−70.3 − 121. i)11-s + 1.11e3·13-s + 835.·15-s + (27.4 + 47.5i)17-s + (−855. + 1.48e3i)19-s + (1.64e3 − 2.84e3i)23-s + (−2.74e3 − 4.75e3i)25-s + 729·27-s − 3.79e3·29-s + (−2.42e3 − 4.19e3i)31-s + (−633. + 1.09e3i)33-s + (5.68e3 − 9.84e3i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.830 + 1.43i)5-s + (−0.166 + 0.288i)9-s + (−0.175 − 0.303i)11-s + 1.82·13-s + 0.958·15-s + (0.0230 + 0.0398i)17-s + (−0.543 + 0.942i)19-s + (0.647 − 1.12i)23-s + (−0.878 − 1.52i)25-s + 0.192·27-s − 0.837·29-s + (−0.452 − 0.784i)31-s + (−0.101 + 0.175i)33-s + (0.682 − 1.18i)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

\(L(3)\) \(\approx\) \(1.482707466\)
\(L(\frac12)\) \(\approx\) \(1.482707466\)
\(L(\frac{7}{2})\) 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 \( 1 + (46.4 - 80.3i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (70.3 + 121. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 1.11e3T + 3.71e5T^{2} \)
17 \( 1 + (-27.4 - 47.5i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (855. - 1.48e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-1.64e3 + 2.84e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 3.79e3T + 2.05e7T^{2} \)
31 \( 1 + (2.42e3 + 4.19e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-5.68e3 + 9.84e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.03e4T + 1.15e8T^{2} \)
43 \( 1 - 7.13e3T + 1.47e8T^{2} \)
47 \( 1 + (8.20e3 - 1.42e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.04e4 - 1.81e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.81e4 + 3.13e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.47e3 + 4.28e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.14e4 - 1.98e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 2.63e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.76e4 - 4.79e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-2.49e4 + 4.32e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 4.48e4T + 3.93e9T^{2} \)
89 \( 1 + (-6.39e4 + 1.10e5i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 6.56e4T + 8.58e9T^{2} \)
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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

−10.47860170879590065496709682270, −9.023465834159755314315616098733, −8.025785941214688908660850533677, −7.42377008553166407080261920481, −6.33472450832149523977959188111, −5.93558755208266795060211851269, −4.14937366059269489125663190653, −3.39228616223431254588499237447, −2.25282808466384982199113150622, −0.76584839327575808548131041259, 0.51533397506853286279368896396, 1.46324361554790387047470707127, 3.38358124510480296654621372773, 4.22867755304245865214447485899, 5.03060154365125062957758293725, 5.91949850669519986425083679232, 7.19210613730961670651184978345, 8.294490047036297988266906198400, 8.854283785954001424562194205455, 9.579949037548859817820316451292

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