Properties

Label 2-2268-1.1-c1-0-17
Degree $2$
Conductor $2268$
Sign $-1$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
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  − 0.239·5-s − 7-s + 5.12·11-s − 4.88·13-s − 3.70·17-s + 3.66·19-s − 7.42·23-s − 4.94·25-s − 3.46·29-s − 0.717·31-s + 0.239·35-s + 4.60·37-s − 5.60·41-s + 12.4·43-s − 4.33·47-s + 49-s + 0.942·53-s − 1.22·55-s − 7.57·59-s + 5.50·61-s + 1.16·65-s − 0.660·67-s − 13.7·71-s − 3.66·73-s − 5.12·77-s − 6.22·79-s − 9.70·83-s + ⋯
L(s)  = 1  − 0.106·5-s − 0.377·7-s + 1.54·11-s − 1.35·13-s − 0.898·17-s + 0.839·19-s − 1.54·23-s − 0.988·25-s − 0.643·29-s − 0.128·31-s + 0.0404·35-s + 0.756·37-s − 0.875·41-s + 1.90·43-s − 0.633·47-s + 0.142·49-s + 0.129·53-s − 0.165·55-s − 0.986·59-s + 0.705·61-s + 0.144·65-s − 0.0806·67-s − 1.63·71-s − 0.428·73-s − 0.584·77-s − 0.700·79-s − 1.06·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-1$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 0.239T + 5T^{2} \)
11 \( 1 - 5.12T + 11T^{2} \)
13 \( 1 + 4.88T + 13T^{2} \)
17 \( 1 + 3.70T + 17T^{2} \)
19 \( 1 - 3.66T + 19T^{2} \)
23 \( 1 + 7.42T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 + 0.717T + 31T^{2} \)
37 \( 1 - 4.60T + 37T^{2} \)
41 \( 1 + 5.60T + 41T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 + 4.33T + 47T^{2} \)
53 \( 1 - 0.942T + 53T^{2} \)
59 \( 1 + 7.57T + 59T^{2} \)
61 \( 1 - 5.50T + 61T^{2} \)
67 \( 1 + 0.660T + 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 + 3.66T + 73T^{2} \)
79 \( 1 + 6.22T + 79T^{2} \)
83 \( 1 + 9.70T + 83T^{2} \)
89 \( 1 - 7.48T + 89T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−8.752857055851804602153299587987, −7.73614878834370079651814229214, −7.13680425738851781114520364224, −6.30293316676175881342753178060, −5.59363285794772898477881406677, −4.39095307747203220138943377344, −3.88200947149418833363244070641, −2.67870240381194052101944133764, −1.63042204979926934887120813994, 0, 1.63042204979926934887120813994, 2.67870240381194052101944133764, 3.88200947149418833363244070641, 4.39095307747203220138943377344, 5.59363285794772898477881406677, 6.30293316676175881342753178060, 7.13680425738851781114520364224, 7.73614878834370079651814229214, 8.752857055851804602153299587987

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