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Properties

Label 2-588-12.11-c1-0-43
Degree $2$
Conductor $588$
Sign $0.0126 - 0.999i$
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  + (1.13 + 0.842i)2-s + (1.65 + 0.522i)3-s + (0.578 + 1.91i)4-s + 0.499i·5-s + (1.43 + 1.98i)6-s + (−0.956 + 2.66i)8-s + (2.45 + 1.72i)9-s + (−0.421 + 0.567i)10-s + 1.39·11-s + (−0.0438 + 3.46i)12-s − 2.75·13-s + (−0.260 + 0.824i)15-s + (−3.32 + 2.21i)16-s − 5.82i·17-s + (1.33 + 4.02i)18-s − 2.48i·19-s + ⋯
L(s)  = 1  + (0.802 + 0.596i)2-s + (0.953 + 0.301i)3-s + (0.289 + 0.957i)4-s + 0.223i·5-s + (0.585 + 0.810i)6-s + (−0.338 + 0.941i)8-s + (0.818 + 0.575i)9-s + (−0.133 + 0.179i)10-s + 0.419·11-s + (−0.0126 + 0.999i)12-s − 0.762·13-s + (−0.0673 + 0.212i)15-s + (−0.832 + 0.554i)16-s − 1.41i·17-s + (0.314 + 0.949i)18-s − 0.569i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0126 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0126 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.0126 - 0.999i$
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: \((2,\ 588,\ (\ :1/2),\ 0.0126 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16450 + 2.13727i\)
\(L(\frac12)\) \(\approx\) \(2.16450 + 2.13727i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 - 0.842i)T \)
3 \( 1 + (-1.65 - 0.522i)T \)
7 \( 1 \)
good5 \( 1 - 0.499iT - 5T^{2} \)
11 \( 1 - 1.39T + 11T^{2} \)
13 \( 1 + 2.75T + 13T^{2} \)
17 \( 1 + 5.82iT - 17T^{2} \)
19 \( 1 + 2.48iT - 19T^{2} \)
23 \( 1 + 5.06T + 23T^{2} \)
29 \( 1 - 6.32iT - 29T^{2} \)
31 \( 1 + 6.91iT - 31T^{2} \)
37 \( 1 - 7.34T + 37T^{2} \)
41 \( 1 - 5.74iT - 41T^{2} \)
43 \( 1 + 3.52iT - 43T^{2} \)
47 \( 1 + 5.06T + 47T^{2} \)
53 \( 1 + 5.24iT - 53T^{2} \)
59 \( 1 + 3.15T + 59T^{2} \)
61 \( 1 - 4.90T + 61T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 + 4.54T + 71T^{2} \)
73 \( 1 - 9.97T + 73T^{2} \)
79 \( 1 + 2.13iT - 79T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 + 2.45iT - 89T^{2} \)
97 \( 1 + 0.526T + 97T^{2} \)
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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

−11.05201625162724480380685802878, −9.784467309247815068154576145908, −9.106119480224993606248673518934, −8.067044720794076827216077766626, −7.31783383408023008782902169798, −6.55133070752119078543813632975, −5.13651632756299600878503258067, −4.39369780102567112140998026442, −3.23443628606715428426720594420, −2.36689995271946033325933825208, 1.44825054827152608074476316123, 2.54878069647133526291931261927, 3.73940330272555424991556435548, 4.49327425916174975917559941246, 5.87214791261380918209723838188, 6.74978660270449598576825326555, 7.87038111554225394244673140016, 8.789839811021393341537843440712, 9.798262199465248933258209142528, 10.35224461115736730278124148390

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