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Properties

Label 2-588-3.2-c0-0-0
Degree $2$
Conductor $588$
Sign $1$
Analytic cond. $0.293450$
Root an. cond. $0.541710$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 13-s + 19-s + 25-s − 27-s + 31-s − 37-s − 39-s − 43-s − 57-s − 2·61-s − 67-s + 73-s − 75-s − 79-s + 81-s − 93-s − 2·97-s + 103-s − 109-s + 111-s + 117-s + ⋯
L(s)  = 1  − 3-s + 9-s + 13-s + 19-s + 25-s − 27-s + 31-s − 37-s − 39-s − 43-s − 57-s − 2·61-s − 67-s + 73-s − 75-s − 79-s + 81-s − 93-s − 2·97-s + 103-s − 109-s + 111-s + 117-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.293450\)
Root analytic conductor: \(0.541710\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{588} (197, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7245109679\)
\(L(\frac12)\) \(\approx\) \(0.7245109679\)
\(L(1)\) 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 \( 1 + T \)
7 \( 1 \)
good5 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 - T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 - T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 - T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 - T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 + T )^{2} \)
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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

−10.91074445526676102112351689082, −10.24659343340798642361025291776, −9.271450018166557252925127806422, −8.239448581511432914659821453158, −7.14372411741639918660033059449, −6.34524131862809207566413238709, −5.43394338359496198936862937802, −4.50756458861580204820438769106, −3.25184872341951126116625108978, −1.34244131093637505982439762316, 1.34244131093637505982439762316, 3.25184872341951126116625108978, 4.50756458861580204820438769106, 5.43394338359496198936862937802, 6.34524131862809207566413238709, 7.14372411741639918660033059449, 8.239448581511432914659821453158, 9.271450018166557252925127806422, 10.24659343340798642361025291776, 10.91074445526676102112351689082

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