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Properties

Label 2-591-591.590-c0-0-9
Degree $2$
Conductor $591$
Sign $1$
Analytic cond. $0.294947$
Root an. cond. $0.543090$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.68·2-s + 3-s + 1.83·4-s − 1.91·5-s + 1.68·6-s − 0.284·7-s + 1.39·8-s + 9-s − 3.22·10-s − 1.30·11-s + 1.83·12-s − 0.478·14-s − 1.91·15-s + 0.521·16-s + 0.830·17-s + 1.68·18-s − 1.30·19-s − 3.51·20-s − 0.284·21-s − 2.20·22-s + 1.39·24-s + 2.68·25-s + 27-s − 0.521·28-s − 3.22·30-s − 0.521·32-s − 1.30·33-s + ⋯
L(s)  = 1  + 1.68·2-s + 3-s + 1.83·4-s − 1.91·5-s + 1.68·6-s − 0.284·7-s + 1.39·8-s + 9-s − 3.22·10-s − 1.30·11-s + 1.83·12-s − 0.478·14-s − 1.91·15-s + 0.521·16-s + 0.830·17-s + 1.68·18-s − 1.30·19-s − 3.51·20-s − 0.284·21-s − 2.20·22-s + 1.39·24-s + 2.68·25-s + 27-s − 0.521·28-s − 3.22·30-s − 0.521·32-s − 1.30·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 591 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 591 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(591\)    =    \(3 \cdot 197\)
Sign: $1$
Analytic conductor: \(0.294947\)
Root analytic conductor: \(0.543090\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{591} (590, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 591,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.120358964\)
\(L(\frac12)\) \(\approx\) \(2.120358964\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
197 \( 1 - T \)
good2 \( 1 - 1.68T + T^{2} \)
5 \( 1 + 1.91T + T^{2} \)
7 \( 1 + 0.284T + T^{2} \)
11 \( 1 + 1.30T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 0.830T + T^{2} \)
19 \( 1 + 1.30T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 0.830T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.68T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.91T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 0.284T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.68T + T^{2} \)
97 \( 1 - 0.830T + T^{2} \)
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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

−11.11819327255249116523739518260, −10.42024625195331145823785549147, −8.886681966505130115516048030272, −7.82063880061890677769549391375, −7.50417240217804859185671175052, −6.31538538301541073400568703538, −4.88055421624727734066880052540, −4.18550701066793006011325118947, −3.37579794647086645013551405443, −2.61041148005977183526182438709, 2.61041148005977183526182438709, 3.37579794647086645013551405443, 4.18550701066793006011325118947, 4.88055421624727734066880052540, 6.31538538301541073400568703538, 7.50417240217804859185671175052, 7.82063880061890677769549391375, 8.886681966505130115516048030272, 10.42024625195331145823785549147, 11.11819327255249116523739518260

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