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Properties

Label 2-588-84.11-c1-0-65
Degree $2$
Conductor $588$
Sign $-0.251 + 0.967i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.938 − 1.05i)2-s + (1.62 − 0.593i)3-s + (−0.238 − 1.98i)4-s + (−0.695 + 0.401i)5-s + (0.899 − 2.27i)6-s + (−2.32 − 1.61i)8-s + (2.29 − 1.93i)9-s + (−0.228 + 1.11i)10-s + (1.17 − 2.03i)11-s + (−1.56 − 3.09i)12-s + 5.26·13-s + (−0.893 + 1.06i)15-s + (−3.88 + 0.945i)16-s + (1.02 + 0.592i)17-s + (0.112 − 4.24i)18-s + (−6.16 + 3.56i)19-s + ⋯
L(s)  = 1  + (0.663 − 0.747i)2-s + (0.939 − 0.342i)3-s + (−0.119 − 0.992i)4-s + (−0.311 + 0.179i)5-s + (0.367 − 0.930i)6-s + (−0.821 − 0.569i)8-s + (0.765 − 0.643i)9-s + (−0.0721 + 0.351i)10-s + (0.353 − 0.612i)11-s + (−0.451 − 0.892i)12-s + 1.46·13-s + (−0.230 + 0.275i)15-s + (−0.971 + 0.236i)16-s + (0.248 + 0.143i)17-s + (0.0265 − 0.999i)18-s + (−1.41 + 0.817i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.251 + 0.967i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.251 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62527 - 2.10132i\)
\(L(\frac12)\) \(\approx\) \(1.62527 - 2.10132i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.938 + 1.05i)T \)
3 \( 1 + (-1.62 + 0.593i)T \)
7 \( 1 \)
good5 \( 1 + (0.695 - 0.401i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.17 + 2.03i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.26T + 13T^{2} \)
17 \( 1 + (-1.02 - 0.592i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.16 - 3.56i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.94 + 6.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.23iT - 29T^{2} \)
31 \( 1 + (-4.24 - 2.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.523 + 0.907i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.16iT - 41T^{2} \)
43 \( 1 - 7.94iT - 43T^{2} \)
47 \( 1 + (-3.04 - 5.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.55 - 4.36i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.331 + 0.573i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.479 + 0.829i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.29 - 4.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.67T + 71T^{2} \)
73 \( 1 + (-0.707 + 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6 + 3.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.18T + 83T^{2} \)
89 \( 1 + (14.1 - 8.18i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.37T + 97T^{2} \)
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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

−10.56970537625914727461181864954, −9.682628568109805208751858166160, −8.598682572149772181040989913801, −8.103084558428227913050532595807, −6.50558532809366529747918323183, −6.05072809592311454841559566203, −4.24604111832381495902593859748, −3.71205125011882756059357733249, −2.57610456416995412898565011414, −1.27386948657189453970095496574, 2.19396101944883221842038471379, 3.70144656045804867133764637811, 4.12380847041115684966191852028, 5.33397265229242883190555653918, 6.52732975362614392424914721504, 7.38612213047602043128843539238, 8.398874201843511663224273578453, 8.777787965964570811392424006536, 9.880852391814741429064308232755, 11.00226254733687148310031198820

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