Properties

Label 2-4704-1.1-c1-0-11
Degree $2$
Conductor $4704$
Sign $1$
Analytic cond. $37.5616$
Root an. cond. $6.12875$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·5-s + 9-s + 6·11-s + 5·13-s + 4·15-s + 2·17-s + 19-s − 6·23-s + 11·25-s − 27-s − 3·31-s − 6·33-s + 3·37-s − 5·39-s − 6·41-s + 5·43-s − 4·45-s − 4·47-s − 2·51-s − 6·53-s − 24·55-s − 57-s − 6·59-s − 2·61-s − 20·65-s + 7·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 1/3·9-s + 1.80·11-s + 1.38·13-s + 1.03·15-s + 0.485·17-s + 0.229·19-s − 1.25·23-s + 11/5·25-s − 0.192·27-s − 0.538·31-s − 1.04·33-s + 0.493·37-s − 0.800·39-s − 0.937·41-s + 0.762·43-s − 0.596·45-s − 0.583·47-s − 0.280·51-s − 0.824·53-s − 3.23·55-s − 0.132·57-s − 0.781·59-s − 0.256·61-s − 2.48·65-s + 0.855·67-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4704,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.210308513\)
\(L(\frac12)\) \(\approx\) \(1.210308513\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.094509694455322443215775248612, −7.74331120372046471144853554512, −6.64544240676434943157541733673, −6.41217664556700363080143259500, −5.34435371931263443494196284815, −4.29781951234858082205765497551, −3.84246195950443913229286198520, −3.35436838889153380127411251331, −1.57593473396747095160810643170, −0.67625968889116693489080910605, 0.67625968889116693489080910605, 1.57593473396747095160810643170, 3.35436838889153380127411251331, 3.84246195950443913229286198520, 4.29781951234858082205765497551, 5.34435371931263443494196284815, 6.41217664556700363080143259500, 6.64544240676434943157541733673, 7.74331120372046471144853554512, 8.094509694455322443215775248612

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