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Properties

Label 2-588-21.2-c2-0-10
Degree $2$
Conductor $588$
Sign $0.903 + 0.428i$
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  + (−2.97 − 0.355i)3-s + (−0.750 − 0.433i)5-s + (8.74 + 2.11i)9-s + (−2.12 + 1.22i)11-s − 6.53·13-s + (2.08 + 1.55i)15-s + (−16.2 + 9.37i)17-s + (6.80 − 11.7i)19-s + (36.3 + 20.9i)23-s + (−12.1 − 20.9i)25-s + (−25.3 − 9.41i)27-s + 22.8i·29-s + (4.24 + 7.34i)31-s + (6.76 − 2.89i)33-s + (20.8 − 36.1i)37-s + ⋯
L(s)  = 1  + (−0.992 − 0.118i)3-s + (−0.150 − 0.0867i)5-s + (0.971 + 0.235i)9-s + (−0.193 + 0.111i)11-s − 0.503·13-s + (0.138 + 0.103i)15-s + (−0.955 + 0.551i)17-s + (0.358 − 0.620i)19-s + (1.58 + 0.912i)23-s + (−0.484 − 0.839i)25-s + (−0.937 − 0.348i)27-s + 0.787i·29-s + (0.136 + 0.237i)31-s + (0.204 − 0.0878i)33-s + (0.564 − 0.977i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.903 + 0.428i$
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: \((2,\ 588,\ (\ :1),\ 0.903 + 0.428i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.077822961\)
\(L(\frac12)\) \(\approx\) \(1.077822961\)
\(L(2)\) 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 \( 1 + (2.97 + 0.355i)T \)
7 \( 1 \)
good5 \( 1 + (0.750 + 0.433i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.12 - 1.22i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 6.53T + 169T^{2} \)
17 \( 1 + (16.2 - 9.37i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.80 + 11.7i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-36.3 - 20.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 22.8iT - 841T^{2} \)
31 \( 1 + (-4.24 - 7.34i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-20.8 + 36.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 71.1iT - 1.68e3T^{2} \)
43 \( 1 - 59.7T + 1.84e3T^{2} \)
47 \( 1 + (-60.4 - 34.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-54.0 + 31.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-49.4 + 28.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-29.6 + 51.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (33 + 57.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 53.7iT - 5.04e3T^{2} \)
73 \( 1 + (-47.7 - 82.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-20.4 + 35.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 23.0iT - 6.88e3T^{2} \)
89 \( 1 + (4.78 + 2.76i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 85.5T + 9.40e3T^{2} \)
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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

−10.73085785343143845914725845123, −9.615151114868357050068237313896, −8.774497972891302375884016080290, −7.44336744055610285755302239911, −6.92041983457027000537360142793, −5.75516638549287342640016291042, −4.95783783661618782674111037362, −3.95653777079762494402243263953, −2.28912506101432221581287778723, −0.68280423404168783519305201224, 0.853141027666047093674185284576, 2.62084570180282552443797205973, 4.14493844091083748399182975831, 4.99298682647089722237355612102, 5.95843973940531680650335623820, 6.91379427007198331277377188498, 7.66633977239136909341430161681, 8.954101156634377204672281264072, 9.812706245379652943571979937807, 10.66023061579006855854093472281

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