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Properties

Label 2-588-7.3-c2-0-9
Degree $2$
Conductor $588$
Sign $0.0956 + 0.995i$
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  + (−1.5 − 0.866i)3-s + (5.04 − 2.91i)5-s + (1.5 + 2.59i)9-s + (3.43 − 5.94i)11-s − 3.62i·13-s − 10.0·15-s + (−8.41 − 4.85i)17-s + (26.2 − 15.1i)19-s + (9.07 + 15.7i)23-s + (4.48 − 7.76i)25-s − 5.19i·27-s − 40.4·29-s + (47.8 + 27.6i)31-s + (−10.2 + 5.94i)33-s + (−27.4 − 47.6i)37-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (1.00 − 0.582i)5-s + (0.166 + 0.288i)9-s + (0.311 − 0.540i)11-s − 0.278i·13-s − 0.672·15-s + (−0.494 − 0.285i)17-s + (1.38 − 0.797i)19-s + (0.394 + 0.683i)23-s + (0.179 − 0.310i)25-s − 0.192i·27-s − 1.39·29-s + (1.54 + 0.890i)31-s + (−0.311 + 0.180i)33-s + (−0.742 − 1.28i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0956 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0956 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.0956 + 0.995i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ 0.0956 + 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.728653548\)
\(L(\frac12)\) \(\approx\) \(1.728653548\)
\(L(2)\) 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 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-5.04 + 2.91i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-3.43 + 5.94i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 3.62iT - 169T^{2} \)
17 \( 1 + (8.41 + 4.85i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-26.2 + 15.1i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-9.07 - 15.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 40.4T + 841T^{2} \)
31 \( 1 + (-47.8 - 27.6i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (27.4 + 47.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 56.3iT - 1.68e3T^{2} \)
43 \( 1 + 66.0T + 1.84e3T^{2} \)
47 \( 1 + (-42.8 + 24.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-40.5 + 70.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (30.1 + 17.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (0.0331 - 0.0191i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-32.0 + 55.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 50.2T + 5.04e3T^{2} \)
73 \( 1 + (18.4 + 10.6i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-23.7 - 41.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 33.6iT - 6.88e3T^{2} \)
89 \( 1 + (-135. + 78.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 43.7iT - 9.40e3T^{2} \)
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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

−10.25527293324885838177082835812, −9.352768075250036627526286337464, −8.750622088089080200525124298562, −7.45539852874459565807518965629, −6.61554910796625818253284244136, −5.48263788930256772670651474684, −5.11664617974680322206396854606, −3.47522462123709215149207505974, −1.98236546268811773733567621780, −0.74176144011634942267149674777, 1.47026523672639611023057157381, 2.79370148311205635093988485199, 4.16311359254561786549863986401, 5.25679160775409889924443242396, 6.20104518480952647123636988260, 6.83208610247546572759109652923, 7.999203859854023074334764335917, 9.293496422396353504277982213243, 9.883858202797611914212262434107, 10.52635630562039624721407364391

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