Properties

Label 2-4600-1.1-c1-0-32
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  + 2·3-s − 7-s + 9-s − 5·11-s − 13-s + 4·17-s + 7·19-s − 2·21-s + 23-s − 4·27-s + 5·29-s + 2·31-s − 10·33-s + 2·37-s − 2·39-s + 11·41-s − 43-s + 8·47-s − 6·49-s + 8·51-s + 14·57-s − 14·59-s + 10·61-s − 63-s + 8·67-s + 2·69-s − 10·71-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.970·17-s + 1.60·19-s − 0.436·21-s + 0.208·23-s − 0.769·27-s + 0.928·29-s + 0.359·31-s − 1.74·33-s + 0.328·37-s − 0.320·39-s + 1.71·41-s − 0.152·43-s + 1.16·47-s − 6/7·49-s + 1.12·51-s + 1.85·57-s − 1.82·59-s + 1.28·61-s − 0.125·63-s + 0.977·67-s + 0.240·69-s − 1.18·71-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: $\chi_{4600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.609397357\)
\(L(\frac12)\) \(\approx\) \(2.609397357\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−8.119890266233448764673073286953, −7.75933913337492956708543983542, −7.23007117008369367255787736573, −6.03994603636818306950512062212, −5.37800953098479694736026160503, −4.58097637085452208340192660052, −3.38235747826954915221899397227, −2.97043737907793024466586140959, −2.26392041373491498443255092067, −0.840764340974826730158997134651, 0.840764340974826730158997134651, 2.26392041373491498443255092067, 2.97043737907793024466586140959, 3.38235747826954915221899397227, 4.58097637085452208340192660052, 5.37800953098479694736026160503, 6.03994603636818306950512062212, 7.23007117008369367255787736573, 7.75933913337492956708543983542, 8.119890266233448764673073286953

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