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Properties

Label 2-588-3.2-c2-0-10
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 − 13-s − 37·19-s + 25·25-s − 27·27-s + 59·31-s + 47·37-s + 3·39-s + 83·43-s + 111·57-s + 74·61-s − 109·67-s + 143·73-s − 75·75-s + 131·79-s + 81·81-s − 177·93-s + 2·97-s − 37·103-s + 143·109-s − 141·111-s − 9·117-s + ⋯
L(s)  = 1  − 3-s + 9-s − 0.0769·13-s − 1.94·19-s + 25-s − 27-s + 1.90·31-s + 1.27·37-s + 1/13·39-s + 1.93·43-s + 1.94·57-s + 1.21·61-s − 1.62·67-s + 1.95·73-s − 75-s + 1.65·79-s + 81-s − 1.90·93-s + 2/97·97-s − 0.359·103-s + 1.31·109-s − 1.27·111-s − 0.0769·117-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

\(L(\frac{3}{2})\) \(\approx\) \(1.168925165\)
\(L(\frac12)\) \(\approx\) \(1.168925165\)
\(L(2)\) 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 \( 1 + T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 + 37 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 - 59 T + p^{2} T^{2} \)
37 \( 1 - 47 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 - 83 T + p^{2} T^{2} \)
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 \( 1 + 109 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 - 143 T + p^{2} T^{2} \)
79 \( 1 - 131 T + p^{2} T^{2} \)
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

−10.64730126826559007306242132702, −9.819625031690945234397679726207, −8.750955768348150984799103319165, −7.74874092365802607969022602278, −6.62355164994974898591822585522, −6.08531144236745047729414475528, −4.85604050535657646192242420622, −4.12077336236658671270546938280, −2.41880050049376289279037620042, −0.799187143364379647241443196618, 0.799187143364379647241443196618, 2.41880050049376289279037620042, 4.12077336236658671270546938280, 4.85604050535657646192242420622, 6.08531144236745047729414475528, 6.62355164994974898591822585522, 7.74874092365802607969022602278, 8.750955768348150984799103319165, 9.819625031690945234397679726207, 10.64730126826559007306242132702

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