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Properties

Label 2-588-21.17-c3-0-28
Degree $2$
Conductor $588$
Sign $0.998 - 0.0579i$
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  + (−1.68 + 4.91i)3-s + (6.50 + 11.2i)5-s + (−21.2 − 16.5i)9-s + (4.49 + 2.59i)11-s − 72.6i·13-s + (−66.3 + 12.9i)15-s + (44.7 − 77.5i)17-s + (113. − 65.2i)19-s + (108. − 62.4i)23-s + (−22.0 + 38.2i)25-s + (117. − 76.6i)27-s − 63.1i·29-s + (−91.6 − 52.9i)31-s + (−20.3 + 17.7i)33-s + (46.9 + 81.2i)37-s + ⋯
L(s)  = 1  + (−0.325 + 0.945i)3-s + (0.581 + 1.00i)5-s + (−0.788 − 0.614i)9-s + (0.123 + 0.0711i)11-s − 1.55i·13-s + (−1.14 + 0.222i)15-s + (0.638 − 1.10i)17-s + (1.36 − 0.788i)19-s + (0.980 − 0.566i)23-s + (−0.176 + 0.305i)25-s + (0.837 − 0.546i)27-s − 0.404i·29-s + (−0.531 − 0.306i)31-s + (−0.107 + 0.0934i)33-s + (0.208 + 0.361i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0579i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.998 - 0.0579i$
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: \((2,\ 588,\ (\ :3/2),\ 0.998 - 0.0579i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.839743042\)
\(L(\frac12)\) \(\approx\) \(1.839743042\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.68 - 4.91i)T \)
7 \( 1 \)
good5 \( 1 + (-6.50 - 11.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-4.49 - 2.59i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 72.6iT - 2.19e3T^{2} \)
17 \( 1 + (-44.7 + 77.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-113. + 65.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-108. + 62.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 63.1iT - 2.43e4T^{2} \)
31 \( 1 + (91.6 + 52.9i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-46.9 - 81.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 320.T + 6.89e4T^{2} \)
43 \( 1 + 351.T + 7.95e4T^{2} \)
47 \( 1 + (-102. - 177. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-68.9 - 39.8i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-449. + 778. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (489. - 282. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-41.1 + 71.2i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 457. iT - 3.57e5T^{2} \)
73 \( 1 + (-769. - 444. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (247. + 429. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.02e3T + 5.71e5T^{2} \)
89 \( 1 + (-65.5 - 113. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 897. iT - 9.12e5T^{2} \)
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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

−10.06887970051229211821619091791, −9.860583036307665682440506449876, −8.740171145598176840013538935030, −7.51712874600808840863450963937, −6.60628370384527246873365249726, −5.52933316706306504272010286768, −4.93477901125904490546962403936, −3.28661011146654622545472400670, −2.82087949926240087839401628659, −0.63816031203290197965875149013, 1.21868654372794091815437841460, 1.77142011781430660198708753070, 3.51770697778163985422074496055, 5.01068552867696348501582388274, 5.66044853037788156836730060784, 6.67080598164687622289127835077, 7.54182975443799216224477547398, 8.599255580730605672261747120065, 9.220788676805339804517334497726, 10.24028819555298713768855130647

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