Properties

Label 2-4600-1.1-c1-0-64
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  − 0.878·3-s − 0.121·7-s − 2.22·9-s + 2.87·11-s − 5.22·13-s + 2.22·17-s + 1.22·19-s + 0.106·21-s − 23-s + 4.59·27-s + 9.34·29-s − 2.12·31-s − 2.52·33-s − 5.59·37-s + 4.59·39-s + 8.22·41-s − 8·43-s + 10.4·47-s − 6.98·49-s − 1.95·51-s + 3.59·53-s − 1.07·57-s − 0.650·59-s − 7.33·61-s + 0.270·63-s − 5.59·67-s + 0.878·69-s + ⋯
L(s)  = 1  − 0.507·3-s − 0.0459·7-s − 0.742·9-s + 0.867·11-s − 1.45·13-s + 0.540·17-s + 0.281·19-s + 0.0232·21-s − 0.208·23-s + 0.883·27-s + 1.73·29-s − 0.381·31-s − 0.440·33-s − 0.919·37-s + 0.735·39-s + 1.28·41-s − 1.21·43-s + 1.52·47-s − 0.997·49-s − 0.274·51-s + 0.493·53-s − 0.142·57-s − 0.0846·59-s − 0.939·61-s + 0.0341·63-s − 0.683·67-s + 0.105·69-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 \( 1 + 0.878T + 3T^{2} \)
7 \( 1 + 0.121T + 7T^{2} \)
11 \( 1 - 2.87T + 11T^{2} \)
13 \( 1 + 5.22T + 13T^{2} \)
17 \( 1 - 2.22T + 17T^{2} \)
19 \( 1 - 1.22T + 19T^{2} \)
29 \( 1 - 9.34T + 29T^{2} \)
31 \( 1 + 2.12T + 31T^{2} \)
37 \( 1 + 5.59T + 37T^{2} \)
41 \( 1 - 8.22T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 - 3.59T + 53T^{2} \)
59 \( 1 + 0.650T + 59T^{2} \)
61 \( 1 + 7.33T + 61T^{2} \)
67 \( 1 + 5.59T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 + 3.51T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + 0.486T + 89T^{2} \)
97 \( 1 + 0.635T + 97T^{2} \)
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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

−7.87514941104700597229788115631, −7.19604543688968685088377875065, −6.45486661363908126629452780591, −5.77425889653177670937104707541, −5.02370206111727381503902487309, −4.35890812596204029786926128855, −3.24817954856466083233044248048, −2.51965338054142476582722345748, −1.24989929006281971613781897180, 0, 1.24989929006281971613781897180, 2.51965338054142476582722345748, 3.24817954856466083233044248048, 4.35890812596204029786926128855, 5.02370206111727381503902487309, 5.77425889653177670937104707541, 6.45486661363908126629452780591, 7.19604543688968685088377875065, 7.87514941104700597229788115631

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