Properties

Label 2-4704-1.1-c1-0-16
Degree $2$
Conductor $4704$
Sign $1$
Analytic cond. $37.5616$
Root an. cond. $6.12875$
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  + 3-s − 1.41·5-s + 9-s − 2.82·11-s − 1.41·13-s − 1.41·15-s + 1.41·17-s + 2.82·23-s − 2.99·25-s + 27-s + 4·31-s − 2.82·33-s + 4·37-s − 1.41·39-s − 1.41·41-s − 5.65·43-s − 1.41·45-s + 12·47-s + 1.41·51-s + 10·53-s + 4.00·55-s − 1.41·61-s + 2.00·65-s − 11.3·67-s + 2.82·69-s − 2.82·71-s + 12.7·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.632·5-s + 0.333·9-s − 0.852·11-s − 0.392·13-s − 0.365·15-s + 0.342·17-s + 0.589·23-s − 0.599·25-s + 0.192·27-s + 0.718·31-s − 0.492·33-s + 0.657·37-s − 0.226·39-s − 0.220·41-s − 0.862·43-s − 0.210·45-s + 1.75·47-s + 0.198·51-s + 1.37·53-s + 0.539·55-s − 0.181·61-s + 0.248·65-s − 1.38·67-s + 0.340·69-s − 0.335·71-s + 1.48·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4704\)    =    \(2^{5} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(37.5616\)
Root analytic conductor: \(6.12875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4704,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.831844333\)
\(L(\frac12)\) \(\approx\) \(1.831844333\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + 1.41T + 5T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 1.41T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 2.82T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 + 9.89T + 97T^{2} \)
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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

−8.201028320239681312408090644340, −7.63834231726358929181922723407, −7.13058362291734954708348950476, −6.13864318122478829251218722700, −5.25688080340081370876587586496, −4.52026288550930244206740740442, −3.69892502752570853074797951614, −2.90001928910427370120779907189, −2.11016648875023256614243724091, −0.71419513450046378514867081922, 0.71419513450046378514867081922, 2.11016648875023256614243724091, 2.90001928910427370120779907189, 3.69892502752570853074797951614, 4.52026288550930244206740740442, 5.25688080340081370876587586496, 6.13864318122478829251218722700, 7.13058362291734954708348950476, 7.63834231726358929181922723407, 8.201028320239681312408090644340

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