Properties

Label 2-4851-1.1-c1-0-46
Degree $2$
Conductor $4851$
Sign $1$
Analytic cond. $38.7354$
Root an. cond. $6.22377$
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  − 2.47·2-s + 4.11·4-s − 2.58·5-s − 5.22·8-s + 6.39·10-s + 11-s + 5.87·13-s + 4.70·16-s + 7.51·17-s + 2.35·19-s − 10.6·20-s − 2.47·22-s − 6.94·23-s + 1.69·25-s − 14.5·26-s + 5.87·29-s + 3.66·31-s − 1.16·32-s − 18.5·34-s + 3.30·37-s − 5.83·38-s + 13.5·40-s + 5.28·41-s + 7.40·43-s + 4.11·44-s + 17.1·46-s + 7.53·47-s + ⋯
L(s)  = 1  − 1.74·2-s + 2.05·4-s − 1.15·5-s − 1.84·8-s + 2.02·10-s + 0.301·11-s + 1.62·13-s + 1.17·16-s + 1.82·17-s + 0.540·19-s − 2.38·20-s − 0.527·22-s − 1.44·23-s + 0.339·25-s − 2.84·26-s + 1.09·29-s + 0.657·31-s − 0.206·32-s − 3.18·34-s + 0.543·37-s − 0.945·38-s + 2.13·40-s + 0.825·41-s + 1.12·43-s + 0.620·44-s + 2.53·46-s + 1.09·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4851\)    =    \(3^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(38.7354\)
Root analytic conductor: \(6.22377\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4851,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8069395759\)
\(L(\frac12)\) \(\approx\) \(0.8069395759\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + 2.47T + 2T^{2} \)
5 \( 1 + 2.58T + 5T^{2} \)
13 \( 1 - 5.87T + 13T^{2} \)
17 \( 1 - 7.51T + 17T^{2} \)
19 \( 1 - 2.35T + 19T^{2} \)
23 \( 1 + 6.94T + 23T^{2} \)
29 \( 1 - 5.87T + 29T^{2} \)
31 \( 1 - 3.66T + 31T^{2} \)
37 \( 1 - 3.30T + 37T^{2} \)
41 \( 1 - 5.28T + 41T^{2} \)
43 \( 1 - 7.40T + 43T^{2} \)
47 \( 1 - 7.53T + 47T^{2} \)
53 \( 1 - 4.22T + 53T^{2} \)
59 \( 1 + 0.926T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + 4.45T + 71T^{2} \)
73 \( 1 - 2.12T + 73T^{2} \)
79 \( 1 + 4.45T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−8.321098273911061139492951626275, −7.68951805629520013587166437526, −7.36023158243649014411385252092, −6.23498923005341219644552925377, −5.81027297059699040529290890965, −4.29103394321137974504671575197, −3.60726508121161199654574817796, −2.69896147832521497530703159070, −1.32841937005603245744345055426, −0.74667800640266865769255345393, 0.74667800640266865769255345393, 1.32841937005603245744345055426, 2.69896147832521497530703159070, 3.60726508121161199654574817796, 4.29103394321137974504671575197, 5.81027297059699040529290890965, 6.23498923005341219644552925377, 7.36023158243649014411385252092, 7.68951805629520013587166437526, 8.321098273911061139492951626275

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