Properties

Label 2-588-21.20-c5-0-6
Degree $2$
Conductor $588$
Sign $-0.879 - 0.476i$
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  + (−5.18 + 14.6i)3-s − 64.3·5-s + (−189. − 152. i)9-s − 593. i·11-s + 1.19e3i·13-s + (333. − 945. i)15-s + 1.37e3·17-s − 142. i·19-s − 3.52e3i·23-s + 1.01e3·25-s + (3.22e3 − 1.98e3i)27-s + 8.11e3i·29-s − 6.76e3i·31-s + (8.72e3 + 3.08e3i)33-s + 1.12e4·37-s + ⋯
L(s)  = 1  + (−0.332 + 0.942i)3-s − 1.15·5-s + (−0.778 − 0.627i)9-s − 1.47i·11-s + 1.96i·13-s + (0.383 − 1.08i)15-s + 1.15·17-s − 0.0904i·19-s − 1.39i·23-s + 0.323·25-s + (0.851 − 0.524i)27-s + 1.79i·29-s − 1.26i·31-s + (1.39 + 0.492i)33-s + 1.34·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.879 - 0.476i$
Analytic conductor: \(94.3056\)
Root analytic conductor: \(9.71111\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :5/2),\ -0.879 - 0.476i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7142402329\)
\(L(\frac12)\) \(\approx\) \(0.7142402329\)
\(L(\frac{7}{2})\) 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 \( 1 + (5.18 - 14.6i)T \)
7 \( 1 \)
good5 \( 1 + 64.3T + 3.12e3T^{2} \)
11 \( 1 + 593. iT - 1.61e5T^{2} \)
13 \( 1 - 1.19e3iT - 3.71e5T^{2} \)
17 \( 1 - 1.37e3T + 1.41e6T^{2} \)
19 \( 1 + 142. iT - 2.47e6T^{2} \)
23 \( 1 + 3.52e3iT - 6.43e6T^{2} \)
29 \( 1 - 8.11e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.76e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.12e4T + 6.93e7T^{2} \)
41 \( 1 + 3.84e3T + 1.15e8T^{2} \)
43 \( 1 - 2.53e3T + 1.47e8T^{2} \)
47 \( 1 - 1.69e4T + 2.29e8T^{2} \)
53 \( 1 - 2.57e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.29e3T + 7.14e8T^{2} \)
61 \( 1 - 1.98e4iT - 8.44e8T^{2} \)
67 \( 1 + 3.66e4T + 1.35e9T^{2} \)
71 \( 1 + 4.52e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.77e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.15e4T + 3.07e9T^{2} \)
83 \( 1 - 9.36e3T + 3.93e9T^{2} \)
89 \( 1 + 9.52e4T + 5.58e9T^{2} \)
97 \( 1 + 1.01e4iT - 8.58e9T^{2} \)
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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

−10.47284929391371657827635194939, −9.253104479469438913432819928420, −8.722610161323167510555355103965, −7.74681671509270633747667352838, −6.57952167964411828884151019986, −5.68492968666244839846993476481, −4.44801480080892962578812148034, −3.88754454322953061673480401091, −2.87794419371521815302938318940, −0.895533319893784803923031286207, 0.22939472956352124922469749346, 1.25138137872761374672836576282, 2.66392143716459905831892226340, 3.75161768839474217656224802171, 5.05784534119355931979492728316, 5.86068476977959723914319515552, 7.16580886746909812128523250124, 7.78351388869942921794635353104, 8.114265246527046968402511748720, 9.686276444719529796139717478616

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