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Properties

Label 2-588-7.2-c5-0-15
Degree $2$
Conductor $588$
Sign $0.991 + 0.126i$
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 + (14.2 − 24.6i)5-s + (−40.5 + 70.1i)9-s + (−212. − 367. i)11-s + 508.·13-s − 255.·15-s + (269. + 467. i)17-s + (−1.30e3 + 2.25e3i)19-s + (−130. + 226. i)23-s + (1.15e3 + 2.00e3i)25-s + 729·27-s + 6.87e3·29-s + (−2.84e3 − 4.92e3i)31-s + (−1.90e3 + 3.30e3i)33-s + (−2.45e3 + 4.25e3i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.254 − 0.440i)5-s + (−0.166 + 0.288i)9-s + (−0.528 − 0.915i)11-s + 0.834·13-s − 0.293·15-s + (0.226 + 0.392i)17-s + (−0.827 + 1.43i)19-s + (−0.0514 + 0.0891i)23-s + (0.370 + 0.642i)25-s + 0.192·27-s + 1.51·29-s + (−0.531 − 0.920i)31-s + (−0.305 + 0.528i)33-s + (−0.294 + 0.510i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
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: \((2,\ 588,\ (\ :5/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.820082744\)
\(L(\frac12)\) \(\approx\) \(1.820082744\)
\(L(\frac{7}{2})\) 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 \( 1 + (-14.2 + 24.6i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (212. + 367. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 508.T + 3.71e5T^{2} \)
17 \( 1 + (-269. - 467. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.30e3 - 2.25e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (130. - 226. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 6.87e3T + 2.05e7T^{2} \)
31 \( 1 + (2.84e3 + 4.92e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (2.45e3 - 4.25e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 5.72e3T + 1.15e8T^{2} \)
43 \( 1 + 1.73e3T + 1.47e8T^{2} \)
47 \( 1 + (-5.07e3 + 8.78e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.55e4 - 2.70e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.94e4 - 3.36e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (6.82e3 - 1.18e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.53e4 - 2.66e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 4.56e4T + 1.80e9T^{2} \)
73 \( 1 + (1.08e4 + 1.88e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.61e4 - 2.79e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 4.66e4T + 3.93e9T^{2} \)
89 \( 1 + (-3.18e4 + 5.52e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.15e5T + 8.58e9T^{2} \)
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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

−10.09946153411825530877561109034, −8.658004138272186411748064485891, −8.361999266497959746342997473491, −7.21734164641702694942047517564, −6.00483429611358479369870261313, −5.66241305276809459412793061647, −4.28878859519690641187778836905, −3.12551797117269570882358658605, −1.74252898555037720374066540637, −0.799949472018642118560549603030, 0.56606536666878688454180978338, 2.15064019129525329310418272736, 3.18540312261477354782218810740, 4.46893756428488041674887058583, 5.18597431620250370995910690607, 6.43347697060738350784100460416, 7.01476631882102888865752676077, 8.318144651664217420308685263858, 9.106243368692240533309244670116, 10.16568717877110038269241668142

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