Properties

Label 2-588-7.2-c5-0-26
Degree $2$
Conductor $588$
Sign $-0.266 + 0.963i$
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 + (35.8 − 62.0i)5-s + (−40.5 + 70.1i)9-s + (283. + 491. i)11-s − 831.·13-s − 644.·15-s + (444. + 769. i)17-s + (1.45e3 − 2.52e3i)19-s + (−1.55e3 + 2.68e3i)23-s + (−1.00e3 − 1.73e3i)25-s + 729·27-s + 8.27e3·29-s + (−3.51e3 − 6.08e3i)31-s + (2.55e3 − 4.42e3i)33-s + (5.07e3 − 8.78e3i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.640 − 1.10i)5-s + (−0.166 + 0.288i)9-s + (0.706 + 1.22i)11-s − 1.36·13-s − 0.739·15-s + (0.372 + 0.645i)17-s + (0.926 − 1.60i)19-s + (−0.611 + 1.05i)23-s + (−0.320 − 0.555i)25-s + 0.192·27-s + 1.82·29-s + (−0.656 − 1.13i)31-s + (0.408 − 0.706i)33-s + (0.608 − 1.05i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(94.3056\)
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: \((2,\ 588,\ (\ :5/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.005064653\)
\(L(\frac12)\) \(\approx\) \(2.005064653\)
\(L(\frac{7}{2})\) 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 \( 1 + (-35.8 + 62.0i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-283. - 491. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 831.T + 3.71e5T^{2} \)
17 \( 1 + (-444. - 769. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.45e3 + 2.52e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.55e3 - 2.68e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 8.27e3T + 2.05e7T^{2} \)
31 \( 1 + (3.51e3 + 6.08e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-5.07e3 + 8.78e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 3.09e3T + 1.15e8T^{2} \)
43 \( 1 - 1.50e4T + 1.47e8T^{2} \)
47 \( 1 + (-9.94e3 + 1.72e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-4.60e3 - 7.97e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (5.15e3 + 8.92e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.12e4 - 1.95e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (3.20e3 + 5.55e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 6.12e4T + 1.80e9T^{2} \)
73 \( 1 + (1.48e4 + 2.57e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-7.81e3 + 1.35e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 1.66e3T + 3.93e9T^{2} \)
89 \( 1 + (-3.79e4 + 6.56e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 9.80e4T + 8.58e9T^{2} \)
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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

−9.480493934571309036468557305834, −9.058516048085197400312148237786, −7.67118886436610157467635254037, −7.10758153998839221385755711697, −5.89751611567900320596305456292, −5.06077940167807153813253860129, −4.29943828324176421824393290194, −2.49632413113200367686310444966, −1.55446426082211897692312589327, −0.52645699331340661634076803105, 1.02062251940713316132239054433, 2.62448035244952068694321690546, 3.33629306565436695452286483610, 4.64378193596545868062640878542, 5.78060334620223628473196694089, 6.39416791247236457151031355143, 7.38246427123479102411588831059, 8.482975213750811931662391057697, 9.613589118638656769429819923226, 10.16950797948930139449756474964

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