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Properties

Label 2-588-21.11-c0-0-0
Degree $2$
Conductor $588$
Sign $0.605 + 0.795i$
Analytic cond. $0.293450$
Root an. cond. $0.541710$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)9-s + 13-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s − 0.999·27-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s − 43-s − 0.999·57-s + (1 + 1.73i)61-s + (0.5 − 0.866i)67-s + (−0.5 + 0.866i)73-s + (0.499 + 0.866i)75-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)9-s + 13-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s − 0.999·27-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s − 43-s − 0.999·57-s + (1 + 1.73i)61-s + (0.5 − 0.866i)67-s + (−0.5 + 0.866i)73-s + (0.499 + 0.866i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(0.293450\)
Root analytic conductor: \(0.541710\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :0),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.052467687\)
\(L(\frac12)\) \(\approx\) \(1.052467687\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−10.96664958049183023270048931401, −9.754835440362029041881672962531, −8.816263112096803931522599085736, −8.233278486413240224214762073617, −7.15992582868078404847623386712, −6.46029390944863672063136976744, −5.40309970220705646956852590940, −3.93255342343252861154852138051, −2.84850640503453405746967262457, −1.47570677446529143342630202656, 2.12038407169083188352969656730, 3.52350657733996887860440063389, 4.25375385406327933710441393164, 5.48119958529953798572455739245, 6.38159846360287234458692031407, 7.81183857045515260677118259474, 8.431297864040461709348895744382, 9.334258234405525076489462619641, 10.13663339387518722274320505706, 10.88614218557306661600797609083

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