Properties

Label 2-588-21.5-c3-0-35
Degree $2$
Conductor $588$
Sign $-0.783 + 0.621i$
Analytic cond. $34.6931$
Root an. cond. $5.89008$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.69 − 4.44i)3-s + (4.87 − 8.44i)5-s + (−12.4 − 23.9i)9-s + (42.9 − 24.8i)11-s + 2.47i·13-s + (−24.3 − 44.3i)15-s + (−24.7 − 42.8i)17-s + (−96.0 − 55.4i)19-s + (149. + 86.4i)23-s + (15.0 + 25.9i)25-s + (−140. − 8.92i)27-s − 134. i·29-s + (−2.18 + 1.26i)31-s + (5.47 − 257. i)33-s + (58.5 − 101. i)37-s + ⋯
L(s)  = 1  + (0.518 − 0.855i)3-s + (0.435 − 0.754i)5-s + (−0.462 − 0.886i)9-s + (1.17 − 0.680i)11-s + 0.0528i·13-s + (−0.419 − 0.763i)15-s + (−0.353 − 0.611i)17-s + (−1.15 − 0.669i)19-s + (1.35 + 0.783i)23-s + (0.120 + 0.207i)25-s + (−0.997 − 0.0636i)27-s − 0.861i·29-s + (−0.0126 + 0.00730i)31-s + (0.0288 − 1.36i)33-s + (0.260 − 0.450i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.783 + 0.621i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ -0.783 + 0.621i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.374081534\)
\(L(\frac12)\) \(\approx\) \(2.374081534\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.69 + 4.44i)T \)
7 \( 1 \)
good5 \( 1 + (-4.87 + 8.44i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-42.9 + 24.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 2.47iT - 2.19e3T^{2} \)
17 \( 1 + (24.7 + 42.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (96.0 + 55.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-149. - 86.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 134. iT - 2.43e4T^{2} \)
31 \( 1 + (2.18 - 1.26i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-58.5 + 101. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 160.T + 6.89e4T^{2} \)
43 \( 1 + 442.T + 7.95e4T^{2} \)
47 \( 1 + (-155. + 269. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (248. - 143. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-276. - 478. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (504. + 291. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-450. - 780. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 984. iT - 3.57e5T^{2} \)
73 \( 1 + (178. - 102. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (321. - 557. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 351.T + 5.71e5T^{2} \)
89 \( 1 + (544. - 942. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.36e3iT - 9.12e5T^{2} \)
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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

−9.508782761619678037879515714475, −9.012705553018102810537755889256, −8.388078609014884067044987128128, −7.15968932795916091115894962813, −6.46421072327443876670341724219, −5.44516004617435178415097328363, −4.18220409763522729862127356519, −2.92771520348601821698128936012, −1.65930137852328461866225886355, −0.64845694764152665843529241660, 1.77712172552547342913689787211, 2.91133566673823036626596740593, 3.99418975644565493106869949119, 4.85170956460634001554704803405, 6.24873516673849129953264909243, 6.89262066445518316338287651206, 8.224342753615305873017192505244, 8.981859673748569145204783728667, 9.794816743655090485588229543575, 10.58210454788824615699066080813

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