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Properties

Label 2-588-12.11-c1-0-0
Degree $2$
Conductor $588$
Sign $-0.365 + 0.930i$
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  + (−1.38 + 0.300i)2-s + (−1.24 + 1.20i)3-s + (1.81 − 0.830i)4-s + 2.72i·5-s + (1.35 − 2.03i)6-s + (−2.26 + 1.69i)8-s + (0.0992 − 2.99i)9-s + (−0.819 − 3.76i)10-s − 2.04·11-s + (−1.26 + 3.22i)12-s − 4.44·13-s + (−3.28 − 3.39i)15-s + (2.62 − 3.02i)16-s + 0.660i·17-s + (0.763 + 4.17i)18-s + 2.93i·19-s + ⋯
L(s)  = 1  + (−0.977 + 0.212i)2-s + (−0.718 + 0.695i)3-s + (0.909 − 0.415i)4-s + 1.21i·5-s + (0.554 − 0.832i)6-s + (−0.800 + 0.598i)8-s + (0.0330 − 0.999i)9-s + (−0.259 − 1.19i)10-s − 0.615·11-s + (−0.365 + 0.930i)12-s − 1.23·13-s + (−0.848 − 0.876i)15-s + (0.655 − 0.755i)16-s + 0.160i·17-s + (0.180 + 0.983i)18-s + 0.674i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.365 + 0.930i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.365 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0383953 - 0.0563039i\)
\(L(\frac12)\) \(\approx\) \(0.0383953 - 0.0563039i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.300i)T \)
3 \( 1 + (1.24 - 1.20i)T \)
7 \( 1 \)
good5 \( 1 - 2.72iT - 5T^{2} \)
11 \( 1 + 2.04T + 11T^{2} \)
13 \( 1 + 4.44T + 13T^{2} \)
17 \( 1 - 0.660iT - 17T^{2} \)
19 \( 1 - 2.93iT - 19T^{2} \)
23 \( 1 + 1.04T + 23T^{2} \)
29 \( 1 + 2.06iT - 29T^{2} \)
31 \( 1 + 3.10iT - 31T^{2} \)
37 \( 1 + 9.52T + 37T^{2} \)
41 \( 1 + 7.85iT - 41T^{2} \)
43 \( 1 + 0.530iT - 43T^{2} \)
47 \( 1 - 1.04T + 47T^{2} \)
53 \( 1 + 10.5iT - 53T^{2} \)
59 \( 1 + 3.48T + 59T^{2} \)
61 \( 1 + 0.198T + 61T^{2} \)
67 \( 1 - 4.76iT - 67T^{2} \)
71 \( 1 - 5.52T + 71T^{2} \)
73 \( 1 + 3.03T + 73T^{2} \)
79 \( 1 + 10.9iT - 79T^{2} \)
83 \( 1 + 9.15T + 83T^{2} \)
89 \( 1 + 0.541iT - 89T^{2} \)
97 \( 1 + 6.91T + 97T^{2} \)
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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

−10.98586313152744901632532813280, −10.17684163073224327898562088731, −10.00208033743460252507705854198, −8.805126321955302199628186853528, −7.61016320954355763036271402908, −6.92798644161360527913833999210, −6.01891823574627699375453263867, −5.10728239145754111479546092564, −3.51078613997240703217387385970, −2.29768818162253906681138516055, 0.05693073463938245812905168297, 1.40537137002520287976653174816, 2.68451929370866970463129043744, 4.70369956293797356484426790199, 5.48313516110303068750584326473, 6.74330307762287188944875448359, 7.52112780814224137091344006540, 8.320345722959466110613711402305, 9.168011748216996655256488808230, 10.10176606997703996062798930194

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