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Properties

Label 2-588-147.137-c0-0-0
Degree $2$
Conductor $588$
Sign $0.747 + 0.664i$
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  + (0.826 − 0.563i)3-s + (−0.988 − 0.149i)7-s + (0.365 − 0.930i)9-s + (1.03 − 1.29i)13-s + (0.988 + 1.71i)19-s + (−0.900 + 0.433i)21-s + (−0.988 + 0.149i)25-s + (−0.222 − 0.974i)27-s + (−0.955 + 1.65i)31-s + (−1.40 + 0.432i)37-s + (0.123 − 1.64i)39-s + (−0.658 + 0.317i)43-s + (0.955 + 0.294i)49-s + (1.78 + 0.858i)57-s + (−0.425 + 0.131i)61-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)3-s + (−0.988 − 0.149i)7-s + (0.365 − 0.930i)9-s + (1.03 − 1.29i)13-s + (0.988 + 1.71i)19-s + (−0.900 + 0.433i)21-s + (−0.988 + 0.149i)25-s + (−0.222 − 0.974i)27-s + (−0.955 + 1.65i)31-s + (−1.40 + 0.432i)37-s + (0.123 − 1.64i)39-s + (−0.658 + 0.317i)43-s + (0.955 + 0.294i)49-s + (1.78 + 0.858i)57-s + (−0.425 + 0.131i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.747 + 0.664i$
Analytic conductor: \(0.293450\)
Root analytic conductor: \(0.541710\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :0),\ 0.747 + 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.103688507\)
\(L(\frac12)\) \(\approx\) \(1.103688507\)
\(L(1)\) 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 \( 1 + (-0.826 + 0.563i)T \)
7 \( 1 + (0.988 + 0.149i)T \)
good5 \( 1 + (0.988 - 0.149i)T^{2} \)
11 \( 1 + (0.733 - 0.680i)T^{2} \)
13 \( 1 + (-1.03 + 1.29i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.0747 - 0.997i)T^{2} \)
19 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.40 - 0.432i)T + (0.826 - 0.563i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (0.658 - 0.317i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.955 - 0.294i)T^{2} \)
53 \( 1 + (-0.826 - 0.563i)T^{2} \)
59 \( 1 + (0.988 + 0.149i)T^{2} \)
61 \( 1 + (0.425 - 0.131i)T + (0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-1.44 + 0.218i)T + (0.955 - 0.294i)T^{2} \)
79 \( 1 + (0.955 + 1.65i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.733 + 0.680i)T^{2} \)
97 \( 1 - 1.24T + T^{2} \)
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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

−10.52690946378409646825227710245, −9.940223902234651432235611204604, −8.966024741901565362504048209400, −8.130284454149512787868117493493, −7.37643067881230184360989632866, −6.34772482385799728347617381128, −5.51539921416181138930919721621, −3.57514782796243959163573400822, −3.28239734147959649140593305471, −1.51278655266175496389788403266, 2.14655873873253757769721274675, 3.38449370565176405325727462572, 4.16217554851176583907011092286, 5.43589810531843183286277780432, 6.62387183834305951477772957372, 7.45379832682050987655875903231, 8.687492831505999117808719766486, 9.276748921734576988174056039974, 9.835386640828693563790628829492, 10.99246476794342574126043665573

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