Properties

Label 2-4851-1.1-c1-0-119
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  + 1.93·2-s + 1.74·4-s + 4.18·5-s − 0.491·8-s + 8.10·10-s + 11-s + 3.17·13-s − 4.44·16-s + 6.85·17-s + 0.318·19-s + 7.31·20-s + 1.93·22-s + 1.87·23-s + 12.5·25-s + 6.14·26-s + 3.17·29-s − 9.23·31-s − 7.61·32-s + 13.2·34-s − 7.55·37-s + 0.616·38-s − 2.06·40-s + 9.36·41-s − 10.8·43-s + 1.74·44-s + 3.62·46-s − 8.06·47-s + ⋯
L(s)  = 1  + 1.36·2-s + 0.872·4-s + 1.87·5-s − 0.173·8-s + 2.56·10-s + 0.301·11-s + 0.880·13-s − 1.11·16-s + 1.66·17-s + 0.0731·19-s + 1.63·20-s + 0.412·22-s + 0.390·23-s + 2.51·25-s + 1.20·26-s + 0.589·29-s − 1.65·31-s − 1.34·32-s + 2.27·34-s − 1.24·37-s + 0.100·38-s − 0.325·40-s + 1.46·41-s − 1.66·43-s + 0.263·44-s + 0.533·46-s − 1.17·47-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: $\chi_{4851} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4851,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.286481360\)
\(L(\frac12)\) \(\approx\) \(6.286481360\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.93T + 2T^{2} \)
5 \( 1 - 4.18T + 5T^{2} \)
13 \( 1 - 3.17T + 13T^{2} \)
17 \( 1 - 6.85T + 17T^{2} \)
19 \( 1 - 0.318T + 19T^{2} \)
23 \( 1 - 1.87T + 23T^{2} \)
29 \( 1 - 3.17T + 29T^{2} \)
31 \( 1 + 9.23T + 31T^{2} \)
37 \( 1 + 7.55T + 37T^{2} \)
41 \( 1 - 9.36T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + 8.06T + 47T^{2} \)
53 \( 1 + 0.508T + 53T^{2} \)
59 \( 1 + 7.04T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 2.66T + 67T^{2} \)
71 \( 1 - 5.01T + 71T^{2} \)
73 \( 1 - 4.82T + 73T^{2} \)
79 \( 1 - 5.01T + 79T^{2} \)
83 \( 1 - 3.52T + 83T^{2} \)
89 \( 1 + 1.74T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−8.344381820414615688529454571197, −7.17095350548050608392784585821, −6.41224019461319466803867347218, −5.94362059574123672349181226193, −5.32835419560641165089922455597, −4.86810786232630713338672926491, −3.59794940284287256307695636419, −3.15125698952253710483500591969, −2.07009135354253863752747439972, −1.27395242845868082239409331326, 1.27395242845868082239409331326, 2.07009135354253863752747439972, 3.15125698952253710483500591969, 3.59794940284287256307695636419, 4.86810786232630713338672926491, 5.32835419560641165089922455597, 5.94362059574123672349181226193, 6.41224019461319466803867347218, 7.17095350548050608392784585821, 8.344381820414615688529454571197

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