Properties

Label 2-4600-1.1-c1-0-92
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4·7-s − 2·9-s − 2·11-s − 7·13-s + 4·17-s − 6·19-s + 4·21-s + 23-s − 5·27-s + 5·29-s + 3·31-s − 2·33-s − 2·37-s − 7·39-s − 9·41-s − 8·43-s + 47-s + 9·49-s + 4·51-s + 6·53-s − 6·57-s − 8·59-s − 10·61-s − 8·63-s − 2·67-s + 69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s − 2/3·9-s − 0.603·11-s − 1.94·13-s + 0.970·17-s − 1.37·19-s + 0.872·21-s + 0.208·23-s − 0.962·27-s + 0.928·29-s + 0.538·31-s − 0.348·33-s − 0.328·37-s − 1.12·39-s − 1.40·41-s − 1.21·43-s + 0.145·47-s + 9/7·49-s + 0.560·51-s + 0.824·53-s − 0.794·57-s − 1.04·59-s − 1.28·61-s − 1.00·63-s − 0.244·67-s + 0.120·69-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: \(1\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
23 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−8.034387861490633451898627491741, −7.51575267109449801241987108949, −6.62872930281327627027285953667, −5.50610552003431010359051451218, −4.96636573458812700033501711884, −4.38605040643404791188637200745, −3.10516585216650234415273937973, −2.44634667518859626375593246595, −1.64047907913666212564911496252, 0, 1.64047907913666212564911496252, 2.44634667518859626375593246595, 3.10516585216650234415273937973, 4.38605040643404791188637200745, 4.96636573458812700033501711884, 5.50610552003431010359051451218, 6.62872930281327627027285953667, 7.51575267109449801241987108949, 8.034387861490633451898627491741

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