Properties

Label 2-4600-1.1-c1-0-11
Degree $2$
Conductor $4600$
Sign $1$
Analytic cond. $36.7311$
Root an. cond. $6.06062$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 3·9-s − 6·11-s + 2·13-s + 3·17-s − 6·19-s − 23-s + 3·29-s − 3·31-s − 37-s + 9·41-s + 8·43-s − 4·47-s − 6·49-s − 53-s + 59-s + 8·61-s + 3·63-s + 7·67-s − 5·71-s + 6·73-s + 6·77-s + 9·81-s + 11·83-s + 4·89-s − 2·91-s − 6·97-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s − 1.80·11-s + 0.554·13-s + 0.727·17-s − 1.37·19-s − 0.208·23-s + 0.557·29-s − 0.538·31-s − 0.164·37-s + 1.40·41-s + 1.21·43-s − 0.583·47-s − 6/7·49-s − 0.137·53-s + 0.130·59-s + 1.02·61-s + 0.377·63-s + 0.855·67-s − 0.593·71-s + 0.702·73-s + 0.683·77-s + 81-s + 1.20·83-s + 0.423·89-s − 0.209·91-s − 0.609·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(36.7311\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.051914083\)
\(L(\frac12)\) \(\approx\) \(1.051914083\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 + T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.131821699878571714817457957038, −7.895907093455660908259346655755, −6.81651891786475834789657050361, −5.98644200654968908267147201428, −5.52851532679300588628541511212, −4.68254936968660354804320686566, −3.65382078628700195903836662624, −2.83193822481131617809599919331, −2.16496033805434674415326045680, −0.53727497379349778489510799132, 0.53727497379349778489510799132, 2.16496033805434674415326045680, 2.83193822481131617809599919331, 3.65382078628700195903836662624, 4.68254936968660354804320686566, 5.52851532679300588628541511212, 5.98644200654968908267147201428, 6.81651891786475834789657050361, 7.895907093455660908259346655755, 8.131821699878571714817457957038

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