Properties

Degree $2$
Conductor $4704$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 0.454·5-s + 9-s − 5.79·11-s − 5.88·13-s + 0.454·15-s + 2.90·17-s + 5.88·19-s − 2.90·23-s − 4.79·25-s + 27-s + 3.54·29-s − 4.33·31-s − 5.79·33-s + 7.70·37-s − 5.88·39-s + 9.58·41-s + 10.7·43-s + 0.454·45-s + 4.90·47-s + 2.90·51-s + 13.1·53-s − 2.63·55-s + 5.88·57-s − 1.79·59-s + 4.67·61-s − 2.67·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.203·5-s + 0.333·9-s − 1.74·11-s − 1.63·13-s + 0.117·15-s + 0.705·17-s + 1.34·19-s − 0.606·23-s − 0.958·25-s + 0.192·27-s + 0.658·29-s − 0.779·31-s − 1.00·33-s + 1.26·37-s − 0.942·39-s + 1.49·41-s + 1.64·43-s + 0.0678·45-s + 0.716·47-s + 0.407·51-s + 1.80·53-s − 0.355·55-s + 0.779·57-s − 0.233·59-s + 0.598·61-s − 0.331·65-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: $1$
Motivic weight: \(1\)
Character: $\chi_{4704} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4704,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.036197641\)
\(L(\frac12)\) \(\approx\) \(2.036197641\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.454T + 5T^{2} \)
11 \( 1 + 5.79T + 11T^{2} \)
13 \( 1 + 5.88T + 13T^{2} \)
17 \( 1 - 2.90T + 17T^{2} \)
19 \( 1 - 5.88T + 19T^{2} \)
23 \( 1 + 2.90T + 23T^{2} \)
29 \( 1 - 3.54T + 29T^{2} \)
31 \( 1 + 4.33T + 31T^{2} \)
37 \( 1 - 7.70T + 37T^{2} \)
41 \( 1 - 9.58T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 - 4.90T + 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 + 1.79T + 59T^{2} \)
61 \( 1 - 4.67T + 61T^{2} \)
67 \( 1 - 7.88T + 67T^{2} \)
71 \( 1 + 0.909T + 71T^{2} \)
73 \( 1 - 5.20T + 73T^{2} \)
79 \( 1 + 2.75T + 79T^{2} \)
83 \( 1 - 9.97T + 83T^{2} \)
89 \( 1 + 4.90T + 89T^{2} \)
97 \( 1 + 5.79T + 97T^{2} \)
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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

−8.042468300297955588302791074540, −7.54845546634070910342607778743, −7.32235997848384174768214349395, −5.88011220658887777047232268547, −5.41572699543653049373586273671, −4.64062627788031100277872170587, −3.68110904109015502664387347680, −2.52004236532419179863488110333, −2.41740677076200962223455075500, −0.74199297978788934558945188527, 0.74199297978788934558945188527, 2.41740677076200962223455075500, 2.52004236532419179863488110333, 3.68110904109015502664387347680, 4.64062627788031100277872170587, 5.41572699543653049373586273671, 5.88011220658887777047232268547, 7.32235997848384174768214349395, 7.54845546634070910342607778743, 8.042468300297955588302791074540

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