Properties

Label 2-592-1.1-c1-0-6
Degree $2$
Conductor $592$
Sign $-1$
Analytic cond. $4.72714$
Root an. cond. $2.17419$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.30·3-s − 2.30·5-s + 2.60·7-s + 7.90·9-s + 2.30·11-s + 1.30·13-s + 7.60·15-s − 6·17-s − 2·19-s − 8.60·21-s − 3.90·23-s + 0.302·25-s − 16.2·27-s − 3.90·29-s + 0.302·31-s − 7.60·33-s − 6·35-s + 37-s − 4.30·39-s + 9.90·41-s − 0.605·43-s − 18.2·45-s − 4.60·47-s − 0.211·49-s + 19.8·51-s − 6·53-s − 5.30·55-s + ⋯
L(s)  = 1  − 1.90·3-s − 1.02·5-s + 0.984·7-s + 2.63·9-s + 0.694·11-s + 0.361·13-s + 1.96·15-s − 1.45·17-s − 0.458·19-s − 1.87·21-s − 0.814·23-s + 0.0605·25-s − 3.11·27-s − 0.725·29-s + 0.0543·31-s − 1.32·33-s − 1.01·35-s + 0.164·37-s − 0.688·39-s + 1.54·41-s − 0.0923·43-s − 2.71·45-s − 0.671·47-s − 0.0301·49-s + 2.77·51-s − 0.824·53-s − 0.715·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-1$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 - T \)
good3 \( 1 + 3.30T + 3T^{2} \)
5 \( 1 + 2.30T + 5T^{2} \)
7 \( 1 - 2.60T + 7T^{2} \)
11 \( 1 - 2.30T + 11T^{2} \)
13 \( 1 - 1.30T + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 3.90T + 23T^{2} \)
29 \( 1 + 3.90T + 29T^{2} \)
31 \( 1 - 0.302T + 31T^{2} \)
41 \( 1 - 9.90T + 41T^{2} \)
43 \( 1 + 0.605T + 43T^{2} \)
47 \( 1 + 4.60T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 10.6T + 59T^{2} \)
61 \( 1 - 7.51T + 61T^{2} \)
67 \( 1 - 3.51T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + 9.11T + 79T^{2} \)
83 \( 1 + 2.78T + 83T^{2} \)
89 \( 1 + 9.21T + 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−10.78571318363562748886443845898, −9.554024967574301012547120432802, −8.316895950722858324716704808305, −7.38395972065637771494682480961, −6.51846431241654044403836264279, −5.69037175417748359391409666486, −4.45570105167610756826297117658, −4.16909170345753399592649857041, −1.59304163975827628846561282588, 0, 1.59304163975827628846561282588, 4.16909170345753399592649857041, 4.45570105167610756826297117658, 5.69037175417748359391409666486, 6.51846431241654044403836264279, 7.38395972065637771494682480961, 8.316895950722858324716704808305, 9.554024967574301012547120432802, 10.78571318363562748886443845898

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