Properties

Label 2-4704-1.1-c1-0-37
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3.04·5-s + 9-s − 3.93·11-s − 4.88·13-s + 3.04·15-s + 5.34·17-s + 2.30·19-s + 7.93·23-s + 4.25·25-s − 27-s + 5.55·29-s + 0.645·31-s + 3.93·33-s + 5.65·37-s + 4.88·39-s − 10.0·41-s − 8.91·43-s − 3.04·45-s + 6.61·47-s − 5.34·51-s + 1.25·53-s + 11.9·55-s − 2.30·57-s − 3.04·59-s + 2.97·61-s + 14.8·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.36·5-s + 0.333·9-s − 1.18·11-s − 1.35·13-s + 0.785·15-s + 1.29·17-s + 0.528·19-s + 1.65·23-s + 0.851·25-s − 0.192·27-s + 1.03·29-s + 0.115·31-s + 0.684·33-s + 0.929·37-s + 0.782·39-s − 1.57·41-s − 1.35·43-s − 0.453·45-s + 0.964·47-s − 0.748·51-s + 0.172·53-s + 1.61·55-s − 0.304·57-s − 0.396·59-s + 0.380·61-s + 1.84·65-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: $-1$
Analytic conductor: \(37.5616\)
Root analytic conductor: \(6.12875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4704} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4704,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.04T + 5T^{2} \)
11 \( 1 + 3.93T + 11T^{2} \)
13 \( 1 + 4.88T + 13T^{2} \)
17 \( 1 - 5.34T + 17T^{2} \)
19 \( 1 - 2.30T + 19T^{2} \)
23 \( 1 - 7.93T + 23T^{2} \)
29 \( 1 - 5.55T + 29T^{2} \)
31 \( 1 - 0.645T + 31T^{2} \)
37 \( 1 - 5.65T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 + 8.91T + 43T^{2} \)
47 \( 1 - 6.61T + 47T^{2} \)
53 \( 1 - 1.25T + 53T^{2} \)
59 \( 1 + 3.04T + 59T^{2} \)
61 \( 1 - 2.97T + 61T^{2} \)
67 \( 1 - 13.5T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 4.67T + 73T^{2} \)
79 \( 1 - 1.05T + 79T^{2} \)
83 \( 1 + 8.60T + 83T^{2} \)
89 \( 1 + 4.85T + 89T^{2} \)
97 \( 1 + 18.3T + 97T^{2} \)
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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

−7.997231193335318260083135938200, −7.16773061387667418433233919469, −6.80368648277030235437331080394, −5.30928772168521623987519838569, −5.19569360051829732152586539045, −4.30073091427530814546530577986, −3.28031497510224586416552499109, −2.65937500656453602128610385283, −1.03031174212665530956703981297, 0, 1.03031174212665530956703981297, 2.65937500656453602128610385283, 3.28031497510224586416552499109, 4.30073091427530814546530577986, 5.19569360051829732152586539045, 5.30928772168521623987519838569, 6.80368648277030235437331080394, 7.16773061387667418433233919469, 7.997231193335318260083135938200

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