Properties

Label 2-4600-1.1-c1-0-61
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.61·3-s + 3.83·7-s + 3.84·9-s − 0.508·11-s + 1.01·13-s + 1.44·17-s − 0.508·19-s + 10.0·21-s − 23-s + 2.22·27-s + 7.51·29-s − 0.439·31-s − 1.32·33-s − 7.02·37-s + 2.64·39-s + 5.47·41-s + 6.72·43-s − 2.64·47-s + 7.72·49-s + 3.78·51-s + 4.77·53-s − 1.32·57-s + 3.85·59-s − 9.05·61-s + 14.7·63-s + 3.45·67-s − 2.61·69-s + ⋯
L(s)  = 1  + 1.51·3-s + 1.45·7-s + 1.28·9-s − 0.153·11-s + 0.280·13-s + 0.350·17-s − 0.116·19-s + 2.19·21-s − 0.208·23-s + 0.427·27-s + 1.39·29-s − 0.0788·31-s − 0.231·33-s − 1.15·37-s + 0.423·39-s + 0.854·41-s + 1.02·43-s − 0.385·47-s + 1.10·49-s + 0.530·51-s + 0.656·53-s − 0.176·57-s + 0.501·59-s − 1.15·61-s + 1.86·63-s + 0.422·67-s − 0.315·69-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: \(0\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.306476877\)
\(L(\frac12)\) \(\approx\) \(4.306476877\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.61T + 3T^{2} \)
7 \( 1 - 3.83T + 7T^{2} \)
11 \( 1 + 0.508T + 11T^{2} \)
13 \( 1 - 1.01T + 13T^{2} \)
17 \( 1 - 1.44T + 17T^{2} \)
19 \( 1 + 0.508T + 19T^{2} \)
29 \( 1 - 7.51T + 29T^{2} \)
31 \( 1 + 0.439T + 31T^{2} \)
37 \( 1 + 7.02T + 37T^{2} \)
41 \( 1 - 5.47T + 41T^{2} \)
43 \( 1 - 6.72T + 43T^{2} \)
47 \( 1 + 2.64T + 47T^{2} \)
53 \( 1 - 4.77T + 53T^{2} \)
59 \( 1 - 3.85T + 59T^{2} \)
61 \( 1 + 9.05T + 61T^{2} \)
67 \( 1 - 3.45T + 67T^{2} \)
71 \( 1 + 2.73T + 71T^{2} \)
73 \( 1 + 9.21T + 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 + 1.40T + 83T^{2} \)
89 \( 1 - 6.77T + 89T^{2} \)
97 \( 1 + 0.313T + 97T^{2} \)
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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

−8.406088974462187578621227341038, −7.74611958764050037260313606124, −7.26711857456712800466684928207, −6.17220039274314420412949616881, −5.19340088928604749446873704806, −4.45867154156994732650905069020, −3.71792069853896557984936132081, −2.79863612313931827073232021774, −2.05576486004353758940028588503, −1.18993679354799318998732468260, 1.18993679354799318998732468260, 2.05576486004353758940028588503, 2.79863612313931827073232021774, 3.71792069853896557984936132081, 4.45867154156994732650905069020, 5.19340088928604749446873704806, 6.17220039274314420412949616881, 7.26711857456712800466684928207, 7.74611958764050037260313606124, 8.406088974462187578621227341038

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