Properties

Label 2-4600-1.1-c1-0-35
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.56·3-s − 5.12·7-s + 3.56·9-s − 4·11-s + 0.561·13-s + 3.12·17-s + 4·19-s + 13.1·21-s − 23-s − 1.43·27-s − 8.56·29-s + 1.43·31-s + 10.2·33-s + 7.12·37-s − 1.43·39-s + 0.561·41-s + 9.12·43-s + 3.68·47-s + 19.2·49-s − 8·51-s + 4.24·53-s − 10.2·57-s − 6.24·59-s + 11.1·61-s − 18.2·63-s − 6.24·67-s + 2.56·69-s + ⋯
L(s)  = 1  − 1.47·3-s − 1.93·7-s + 1.18·9-s − 1.20·11-s + 0.155·13-s + 0.757·17-s + 0.917·19-s + 2.86·21-s − 0.208·23-s − 0.276·27-s − 1.58·29-s + 0.258·31-s + 1.78·33-s + 1.17·37-s − 0.230·39-s + 0.0876·41-s + 1.39·43-s + 0.537·47-s + 2.74·49-s − 1.12·51-s + 0.583·53-s − 1.35·57-s − 0.813·59-s + 1.42·61-s − 2.29·63-s − 0.763·67-s + 0.308·69-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: Trivial
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 \( 1 + 2.56T + 3T^{2} \)
7 \( 1 + 5.12T + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 0.561T + 13T^{2} \)
17 \( 1 - 3.12T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
29 \( 1 + 8.56T + 29T^{2} \)
31 \( 1 - 1.43T + 31T^{2} \)
37 \( 1 - 7.12T + 37T^{2} \)
41 \( 1 - 0.561T + 41T^{2} \)
43 \( 1 - 9.12T + 43T^{2} \)
47 \( 1 - 3.68T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 + 6.24T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 + 6.24T + 67T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 + 16.5T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−7.50163501992219810866232658636, −7.29287365126195095101783479397, −6.19122537890131129570406385135, −5.83575618830576951362275515074, −5.34883898806530265468696558106, −4.26072686023711780495047812552, −3.34879110571241952300008360331, −2.57032539003755719809084489101, −0.891705102404798781509687668570, 0, 0.891705102404798781509687668570, 2.57032539003755719809084489101, 3.34879110571241952300008360331, 4.26072686023711780495047812552, 5.34883898806530265468696558106, 5.83575618830576951362275515074, 6.19122537890131129570406385135, 7.29287365126195095101783479397, 7.50163501992219810866232658636

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