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Properties

Label 2-588-21.2-c2-0-2
Degree $2$
Conductor $588$
Sign $-0.683 - 0.729i$
Analytic cond. $16.0218$
Root an. cond. $4.00272$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.81 + 1.04i)3-s + (−5.12 − 2.95i)5-s + (6.81 − 5.88i)9-s + (5.12 − 2.95i)11-s + 6·13-s + (17.5 + 2.95i)15-s + (−5.12 + 2.95i)17-s + (11.5 − 19.9i)19-s + (−35.8 − 20.7i)23-s + (4.99 + 8.66i)25-s + (−13 + 23.6i)27-s + 47.3i·29-s + (19.5 + 33.7i)31-s + (−11.3 + 13.6i)33-s + (−23.5 + 40.7i)37-s + ⋯
L(s)  = 1  + (−0.937 + 0.348i)3-s + (−1.02 − 0.591i)5-s + (0.756 − 0.653i)9-s + (0.465 − 0.268i)11-s + 0.461·13-s + (1.16 + 0.197i)15-s + (−0.301 + 0.174i)17-s + (0.605 − 1.04i)19-s + (−1.55 − 0.900i)23-s + (0.199 + 0.346i)25-s + (−0.481 + 0.876i)27-s + 1.63i·29-s + (0.629 + 1.08i)31-s + (−0.342 + 0.414i)33-s + (−0.635 + 1.10i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.683 - 0.729i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ -0.683 - 0.729i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2700160183\)
\(L(\frac12)\) \(\approx\) \(0.2700160183\)
\(L(2)\) 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 \( 1 + (2.81 - 1.04i)T \)
7 \( 1 \)
good5 \( 1 + (5.12 + 2.95i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-5.12 + 2.95i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 6T + 169T^{2} \)
17 \( 1 + (5.12 - 2.95i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-11.5 + 19.9i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (35.8 + 20.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 47.3iT - 841T^{2} \)
31 \( 1 + (-19.5 - 33.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (23.5 - 40.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 22T + 1.84e3T^{2} \)
47 \( 1 + (46.1 + 26.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-46.1 + 26.6i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (87.0 - 50.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (40.5 - 70.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (15.5 + 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 94.6iT - 5.04e3T^{2} \)
73 \( 1 + (8.5 + 14.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-4.5 + 7.79i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 47.3iT - 6.88e3T^{2} \)
89 \( 1 + (-76.8 - 44.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 82T + 9.40e3T^{2} \)
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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

−10.86962497698848168876264991030, −10.14576871174032218382912824579, −8.943554272042808484886639722170, −8.331740255097542580948557050795, −7.07497941314862003889659142676, −6.31390814451827902574433775032, −5.09346122125471116080867961590, −4.37362653371881428384789024174, −3.39092998852905982740848250896, −1.18443649814764976381935039748, 0.13341315718010611665285265017, 1.81937743758101793012904866328, 3.60652706726512255543053252455, 4.39038580048092957060776902091, 5.79591704459783425249376408265, 6.44682811464653858815165561741, 7.65744879289884952981991118340, 7.88501559925908440064883652471, 9.512804436589469083019606206898, 10.30544144779676820023279059828

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