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Properties

Label 2-588-3.2-c2-0-17
Degree $2$
Conductor $588$
Sign $1$
Analytic cond. $16.0218$
Root an. cond. $4.00272$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 9·9-s + 22·13-s − 26·19-s + 25·25-s + 27·27-s + 46·31-s + 26·37-s + 66·39-s − 22·43-s − 78·57-s − 74·61-s + 122·67-s + 46·73-s + 75·75-s − 142·79-s + 81·81-s + 138·93-s − 2·97-s − 194·103-s − 214·109-s + 78·111-s + 198·117-s + ⋯
L(s)  = 1  + 3-s + 9-s + 1.69·13-s − 1.36·19-s + 25-s + 27-s + 1.48·31-s + 0.702·37-s + 1.69·39-s − 0.511·43-s − 1.36·57-s − 1.21·61-s + 1.82·67-s + 0.630·73-s + 75-s − 1.79·79-s + 81-s + 1.48·93-s − 0.0206·97-s − 1.88·103-s − 1.96·109-s + 0.702·111-s + 1.69·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{588} (197, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.802175494\)
\(L(\frac12)\) \(\approx\) \(2.802175494\)
\(L(2)\) 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 \( 1 - p T \)
7 \( 1 \)
good5 \( ( 1 - p T )( 1 + p T ) \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( 1 - 22 T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 + 26 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 - 46 T + p^{2} T^{2} \)
37 \( 1 - 26 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 + 22 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( ( 1 - p T )( 1 + p T ) \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 + 74 T + p^{2} T^{2} \)
67 \( 1 - 122 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 - 46 T + p^{2} T^{2} \)
79 \( 1 + 142 T + p^{2} T^{2} \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 + 2 T + p^{2} T^{2} \)
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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

−10.47314044260572014851744881508, −9.490650459978150603891130586131, −8.509788163109067965132250832470, −8.226546792188027682394741067841, −6.88450922227002105888489215426, −6.14104446515822204989294667979, −4.61493328647923215596352855244, −3.72323533111995799968497717184, −2.62356125762572730489849657297, −1.26331373468412367309976499560, 1.26331373468412367309976499560, 2.62356125762572730489849657297, 3.72323533111995799968497717184, 4.61493328647923215596352855244, 6.14104446515822204989294667979, 6.88450922227002105888489215426, 8.226546792188027682394741067841, 8.509788163109067965132250832470, 9.490650459978150603891130586131, 10.47314044260572014851744881508

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