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Properties

Label 2-588-7.5-c2-0-9
Degree $2$
Conductor $588$
Sign $0.134 + 0.990i$
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 + (−0.416 − 0.240i)5-s + (1.5 − 2.59i)9-s + (−2.88 − 4.99i)11-s − 1.41i·13-s − 0.832·15-s + (19.9 − 11.5i)17-s + (−9.24 − 5.33i)19-s + (3.29 − 5.71i)23-s + (−12.3 − 21.4i)25-s − 5.19i·27-s − 6.20·29-s + (36.3 − 20.9i)31-s + (−8.64 − 4.99i)33-s + (30.0 − 52.0i)37-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (−0.0832 − 0.0480i)5-s + (0.166 − 0.288i)9-s + (−0.261 − 0.453i)11-s − 0.109i·13-s − 0.0555·15-s + (1.17 − 0.678i)17-s + (−0.486 − 0.280i)19-s + (0.143 − 0.248i)23-s + (−0.495 − 0.858i)25-s − 0.192i·27-s − 0.213·29-s + (1.17 − 0.676i)31-s + (−0.261 − 0.151i)33-s + (0.812 − 1.40i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.134 + 0.990i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ 0.134 + 0.990i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.875543054\)
\(L(\frac12)\) \(\approx\) \(1.875543054\)
\(L(2)\) 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 \( 1 + (0.416 + 0.240i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.88 + 4.99i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 1.41iT - 169T^{2} \)
17 \( 1 + (-19.9 + 11.5i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (9.24 + 5.33i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.29 + 5.71i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 6.20T + 841T^{2} \)
31 \( 1 + (-36.3 + 20.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-30.0 + 52.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 48.8iT - 1.68e3T^{2} \)
43 \( 1 + 51.5T + 1.84e3T^{2} \)
47 \( 1 + (-16.6 - 9.62i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (41.0 + 71.1i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-80.1 + 46.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (4.32 + 2.49i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-1.10 - 1.91i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 80.5T + 5.04e3T^{2} \)
73 \( 1 + (12.0 - 6.95i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-32.4 + 56.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 118. iT - 6.88e3T^{2} \)
89 \( 1 + (-90.2 - 52.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 31.7iT - 9.40e3T^{2} \)
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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

−10.12672428752251776819964275371, −9.463520151983039307947075534707, −8.320906196206933445301951951041, −7.84231746366690133807481036194, −6.73206142469974236840291639275, −5.77041225463295462960012972775, −4.59339771992699003488276137332, −3.38209371828207120255300822729, −2.33963060499990662350437655533, −0.69197144449130572687158007585, 1.54462157828898020084995943894, 2.95433319662176320087477336734, 3.96522145142532908402074937340, 5.04900199711849859279589711672, 6.11998788171742214752889002132, 7.30311472009411746417116959282, 8.065872086162864470614597924885, 8.908115107193835873525294207800, 9.965508211468323418600136660954, 10.39200325285667703866364430193

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