Properties

Label 2-588-147.107-c0-0-0
Degree $2$
Conductor $588$
Sign $0.325 + 0.945i$
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  + (−0.988 − 0.149i)3-s + (−0.733 − 0.680i)7-s + (0.955 + 0.294i)9-s + (0.440 − 1.92i)13-s + (0.733 − 1.26i)19-s + (0.623 + 0.781i)21-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)27-s + (−0.0747 − 0.129i)31-s + (0.123 − 1.64i)37-s + (−0.722 + 1.84i)39-s + (1.19 + 1.49i)43-s + (0.0747 + 0.997i)49-s + (−0.914 + 1.14i)57-s + (−0.134 + 1.79i)61-s + ⋯
L(s)  = 1  + (−0.988 − 0.149i)3-s + (−0.733 − 0.680i)7-s + (0.955 + 0.294i)9-s + (0.440 − 1.92i)13-s + (0.733 − 1.26i)19-s + (0.623 + 0.781i)21-s + (−0.733 + 0.680i)25-s + (−0.900 − 0.433i)27-s + (−0.0747 − 0.129i)31-s + (0.123 − 1.64i)37-s + (−0.722 + 1.84i)39-s + (1.19 + 1.49i)43-s + (0.0747 + 0.997i)49-s + (−0.914 + 1.14i)57-s + (−0.134 + 1.79i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.325 + 0.945i$
Analytic conductor: \(0.293450\)
Root analytic conductor: \(0.541710\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :0),\ 0.325 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6039969768\)
\(L(\frac12)\) \(\approx\) \(0.6039969768\)
\(L(1)\) 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 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (0.733 + 0.680i)T \)
good5 \( 1 + (0.733 - 0.680i)T^{2} \)
11 \( 1 + (-0.826 + 0.563i)T^{2} \)
13 \( 1 + (-0.440 + 1.92i)T + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + (-0.365 - 0.930i)T^{2} \)
19 \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.365 + 0.930i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.123 + 1.64i)T + (-0.988 - 0.149i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (-1.19 - 1.49i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.0747 - 0.997i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (0.134 - 1.79i)T + (-0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.365 + 0.632i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (1.21 - 1.12i)T + (0.0747 - 0.997i)T^{2} \)
79 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.826 - 0.563i)T^{2} \)
97 \( 1 + 0.445T + T^{2} \)
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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

−10.78633089672602809975906214180, −10.06400492228243681979513855961, −9.201820897671545335986412149331, −7.72971333118621795649014534876, −7.21533881119041739788078534254, −6.04364205242385466211708240671, −5.42272541426778125314327821034, −4.17060743136478883305388648157, −2.97447454796092782650989884802, −0.850387021171205720859953640240, 1.81313514156178756525041020053, 3.58989539608701462091752793780, 4.58053483721138891425619758812, 5.80893858356009987357686940213, 6.35843250444775185597883629755, 7.27387488018695372656256939126, 8.611254634631003346435018234611, 9.542405071276280526906127657123, 10.10873822640107706742314434413, 11.23882470857432918760872504502

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