Properties

Label 2-4600-1.1-c1-0-67
Degree $2$
Conductor $4600$
Sign $-1$
Analytic cond. $36.7311$
Root an. cond. $6.06062$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.07·3-s + 2.07·7-s + 6.48·9-s + 5.07·11-s + 3.48·13-s − 6.48·17-s − 7.48·19-s − 6.40·21-s − 23-s − 10.7·27-s − 1.56·29-s + 0.0791·31-s − 15.6·33-s + 9.71·37-s − 10.7·39-s − 0.480·41-s − 8·43-s − 6.96·47-s − 2.67·49-s + 19.9·51-s − 11.7·53-s + 23.0·57-s − 11.5·59-s + 7.88·61-s + 13.4·63-s + 9.71·67-s + 3.07·69-s + ⋯
L(s)  = 1  − 1.77·3-s + 0.785·7-s + 2.16·9-s + 1.53·11-s + 0.965·13-s − 1.57·17-s − 1.71·19-s − 1.39·21-s − 0.208·23-s − 2.06·27-s − 0.289·29-s + 0.0142·31-s − 2.72·33-s + 1.59·37-s − 1.71·39-s − 0.0751·41-s − 1.21·43-s − 1.01·47-s − 0.382·49-s + 2.79·51-s − 1.60·53-s + 3.05·57-s − 1.50·59-s + 1.00·61-s + 1.69·63-s + 1.18·67-s + 0.370·69-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: no
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 \( 1 + 3.07T + 3T^{2} \)
7 \( 1 - 2.07T + 7T^{2} \)
11 \( 1 - 5.07T + 11T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 + 6.48T + 17T^{2} \)
19 \( 1 + 7.48T + 19T^{2} \)
29 \( 1 + 1.56T + 29T^{2} \)
31 \( 1 - 0.0791T + 31T^{2} \)
37 \( 1 - 9.71T + 37T^{2} \)
41 \( 1 + 0.480T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 6.96T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 - 7.88T + 61T^{2} \)
67 \( 1 - 9.71T + 67T^{2} \)
71 \( 1 + 9.67T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 4.59T + 83T^{2} \)
89 \( 1 - 8.31T + 89T^{2} \)
97 \( 1 + 7.23T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.948395915559373590728521199087, −6.74815194530357467066746245574, −6.43271059024484446726529033762, −6.03049509525168221404093351821, −4.85190039839795922709094010303, −4.45661255648631534006281559936, −3.77347063404550293108797487320, −1.96450535390817251664402918682, −1.27600359040301375644315114772, 0, 1.27600359040301375644315114772, 1.96450535390817251664402918682, 3.77347063404550293108797487320, 4.45661255648631534006281559936, 4.85190039839795922709094010303, 6.03049509525168221404093351821, 6.43271059024484446726529033762, 6.74815194530357467066746245574, 7.948395915559373590728521199087

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