Properties

Label 2-588-21.17-c3-0-2
Degree $2$
Conductor $588$
Sign $0.895 - 0.445i$
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  + (−2.60 − 4.49i)3-s + (−8.45 − 14.6i)5-s + (−13.4 + 23.4i)9-s + (−45.7 − 26.4i)11-s + 49.8i·13-s + (−43.7 + 76.1i)15-s + (−0.687 + 1.19i)17-s + (62.9 − 36.3i)19-s + (−166. + 96.4i)23-s + (−80.4 + 139. i)25-s + (140. − 0.841i)27-s + 34.2i·29-s + (81.2 + 46.8i)31-s + (0.549 + 274. i)33-s + (−150. − 260. i)37-s + ⋯
L(s)  = 1  + (−0.501 − 0.865i)3-s + (−0.756 − 1.30i)5-s + (−0.496 + 0.868i)9-s + (−1.25 − 0.724i)11-s + 1.06i·13-s + (−0.753 + 1.31i)15-s + (−0.00980 + 0.0169i)17-s + (0.759 − 0.438i)19-s + (−1.51 + 0.874i)23-s + (−0.643 + 1.11i)25-s + (0.999 − 0.00599i)27-s + 0.219i·29-s + (0.470 + 0.271i)31-s + (0.00289 + 1.44i)33-s + (−0.667 − 1.15i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.895 - 0.445i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ 0.895 - 0.445i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4167766191\)
\(L(\frac12)\) \(\approx\) \(0.4167766191\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.60 + 4.49i)T \)
7 \( 1 \)
good5 \( 1 + (8.45 + 14.6i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (45.7 + 26.4i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 49.8iT - 2.19e3T^{2} \)
17 \( 1 + (0.687 - 1.19i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-62.9 + 36.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (166. - 96.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 34.2iT - 2.43e4T^{2} \)
31 \( 1 + (-81.2 - 46.8i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (150. + 260. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 88.2T + 6.89e4T^{2} \)
43 \( 1 - 136.T + 7.95e4T^{2} \)
47 \( 1 + (283. + 490. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-395. - 228. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-126. + 219. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-215. + 124. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-54.3 + 94.0i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 21.4iT - 3.57e5T^{2} \)
73 \( 1 + (-310. - 179. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-597. - 1.03e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 834.T + 5.71e5T^{2} \)
89 \( 1 + (114. + 199. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.14e3iT - 9.12e5T^{2} \)
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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

−10.53752538100237445577193975409, −9.286310730260612886956396644709, −8.336703738551843345350633565142, −7.84163786475809463183476453727, −6.86342570367146475353115060347, −5.57820238856737231425860751457, −5.03030314581641765836362199244, −3.76948539684692706291147090578, −2.12175220834669884314118067513, −0.814850654763183821912687837467, 0.18156428564882735174228685136, 2.63576121239269458569383000619, 3.46583852480576973123342091650, 4.55506708105514727251829781257, 5.59116343193401678053699490480, 6.55160747016438890153482607347, 7.64244234541269630414677027887, 8.237716025004515623991425778776, 9.876147417156753594652062254360, 10.25116970439169735178963630641

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