Properties

Label 2-4600-1.1-c1-0-8
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  − 2.21·3-s + 2.22·7-s + 1.88·9-s − 1.57·11-s − 3.96·13-s − 0.294·17-s − 7.76·19-s − 4.91·21-s + 23-s + 2.46·27-s − 9.29·29-s + 9.18·31-s + 3.47·33-s + 10.5·37-s + 8.75·39-s − 2.34·41-s − 6.67·43-s + 1.38·47-s − 2.04·49-s + 0.651·51-s + 11.0·53-s + 17.1·57-s − 5.09·59-s − 8.91·61-s + 4.19·63-s + 1.12·67-s − 2.21·69-s + ⋯
L(s)  = 1  − 1.27·3-s + 0.840·7-s + 0.628·9-s − 0.474·11-s − 1.09·13-s − 0.0715·17-s − 1.78·19-s − 1.07·21-s + 0.208·23-s + 0.473·27-s − 1.72·29-s + 1.65·31-s + 0.605·33-s + 1.73·37-s + 1.40·39-s − 0.365·41-s − 1.01·43-s + 0.201·47-s − 0.292·49-s + 0.0912·51-s + 1.51·53-s + 2.27·57-s − 0.663·59-s − 1.14·61-s + 0.528·63-s + 0.136·67-s − 0.266·69-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

\(L(1)\) \(\approx\) \(0.7729079246\)
\(L(\frac12)\) \(\approx\) \(0.7729079246\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.21T + 3T^{2} \)
7 \( 1 - 2.22T + 7T^{2} \)
11 \( 1 + 1.57T + 11T^{2} \)
13 \( 1 + 3.96T + 13T^{2} \)
17 \( 1 + 0.294T + 17T^{2} \)
19 \( 1 + 7.76T + 19T^{2} \)
29 \( 1 + 9.29T + 29T^{2} \)
31 \( 1 - 9.18T + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 + 2.34T + 41T^{2} \)
43 \( 1 + 6.67T + 43T^{2} \)
47 \( 1 - 1.38T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 + 5.09T + 59T^{2} \)
61 \( 1 + 8.91T + 61T^{2} \)
67 \( 1 - 1.12T + 67T^{2} \)
71 \( 1 - 7.60T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 - 0.199T + 97T^{2} \)
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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

−8.132537290584523911439248277487, −7.63726204067998820856516176357, −6.66975952346134145839811713094, −6.14765650565186880907604977999, −5.28043915548220635576116958398, −4.78018421181576912275044699765, −4.14640688511784160713609313731, −2.71128488763579903998468761077, −1.85680878524738529467615954208, −0.50824252168474403704510750921, 0.50824252168474403704510750921, 1.85680878524738529467615954208, 2.71128488763579903998468761077, 4.14640688511784160713609313731, 4.78018421181576912275044699765, 5.28043915548220635576116958398, 6.14765650565186880907604977999, 6.66975952346134145839811713094, 7.63726204067998820856516176357, 8.132537290584523911439248277487

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