Properties

Label 2-4600-1.1-c1-0-73
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  + 0.189·3-s − 1.65·7-s − 2.96·9-s + 0.0759·11-s − 1.24·13-s + 5.17·17-s + 0.792·19-s − 0.313·21-s − 23-s − 1.12·27-s − 2.85·29-s + 8.90·31-s + 0.0143·33-s + 6.92·37-s − 0.235·39-s + 4.64·41-s − 1.75·43-s + 10.0·47-s − 4.25·49-s + 0.980·51-s − 11.1·53-s + 0.150·57-s − 12.4·59-s − 12.0·61-s + 4.91·63-s − 6.96·67-s − 0.189·69-s + ⋯
L(s)  = 1  + 0.109·3-s − 0.626·7-s − 0.988·9-s + 0.0228·11-s − 0.344·13-s + 1.25·17-s + 0.181·19-s − 0.0684·21-s − 0.208·23-s − 0.217·27-s − 0.529·29-s + 1.59·31-s + 0.00250·33-s + 1.13·37-s − 0.0376·39-s + 0.725·41-s − 0.267·43-s + 1.45·47-s − 0.607·49-s + 0.137·51-s − 1.52·53-s + 0.0198·57-s − 1.62·59-s − 1.53·61-s + 0.618·63-s − 0.850·67-s − 0.0227·69-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 \( 1 - 0.189T + 3T^{2} \)
7 \( 1 + 1.65T + 7T^{2} \)
11 \( 1 - 0.0759T + 11T^{2} \)
13 \( 1 + 1.24T + 13T^{2} \)
17 \( 1 - 5.17T + 17T^{2} \)
19 \( 1 - 0.792T + 19T^{2} \)
29 \( 1 + 2.85T + 29T^{2} \)
31 \( 1 - 8.90T + 31T^{2} \)
37 \( 1 - 6.92T + 37T^{2} \)
41 \( 1 - 4.64T + 41T^{2} \)
43 \( 1 + 1.75T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 + 6.96T + 67T^{2} \)
71 \( 1 + 7.71T + 71T^{2} \)
73 \( 1 + 7.49T + 73T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 - 6.68T + 83T^{2} \)
89 \( 1 + 3.03T + 89T^{2} \)
97 \( 1 + 3.21T + 97T^{2} \)
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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

−7.80816151832521605201681421906, −7.48677221916982448718102501076, −6.17248247325326419853328191407, −6.03900913795832432923994858754, −5.02308577867296045845437443497, −4.16839834395885472374634820473, −3.08080063616651354388454507220, −2.75457097100686176368292047757, −1.32324582688478178892996734115, 0, 1.32324582688478178892996734115, 2.75457097100686176368292047757, 3.08080063616651354388454507220, 4.16839834395885472374634820473, 5.02308577867296045845437443497, 6.03900913795832432923994858754, 6.17248247325326419853328191407, 7.48677221916982448718102501076, 7.80816151832521605201681421906

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