Properties

Label 2-2268-21.17-c1-0-9
Degree $2$
Conductor $2268$
Sign $-0.0987 - 0.995i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.09 + 1.89i)5-s + (−1.38 + 2.25i)7-s + (1.26 + 0.732i)11-s + 3.38i·13-s + (1.32 − 2.28i)17-s + (6.87 − 3.97i)19-s + (3.47 − 2.00i)23-s + (0.117 − 0.203i)25-s + 7.75i·29-s + (−0.612 − 0.353i)31-s + (−5.77 − 0.160i)35-s + (1.41 + 2.45i)37-s − 7.48·41-s + 2.54·43-s + (6.27 + 10.8i)47-s + ⋯
L(s)  = 1  + (0.488 + 0.845i)5-s + (−0.523 + 0.851i)7-s + (0.382 + 0.220i)11-s + 0.937i·13-s + (0.320 − 0.555i)17-s + (1.57 − 0.911i)19-s + (0.724 − 0.418i)23-s + (0.0234 − 0.0406i)25-s + 1.43i·29-s + (−0.109 − 0.0634i)31-s + (−0.975 − 0.0271i)35-s + (0.233 + 0.403i)37-s − 1.16·41-s + 0.387·43-s + (0.915 + 1.58i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0987 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0987 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.0987 - 0.995i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.0987 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.844855651\)
\(L(\frac12)\) \(\approx\) \(1.844855651\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.38 - 2.25i)T \)
good5 \( 1 + (-1.09 - 1.89i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.26 - 0.732i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.38iT - 13T^{2} \)
17 \( 1 + (-1.32 + 2.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.87 + 3.97i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.47 + 2.00i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.75iT - 29T^{2} \)
31 \( 1 + (0.612 + 0.353i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.41 - 2.45i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.48T + 41T^{2} \)
43 \( 1 - 2.54T + 43T^{2} \)
47 \( 1 + (-6.27 - 10.8i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.41 + 1.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.71 - 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.75 - 3.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.92 - 5.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (3.95 + 2.28i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.69 + 8.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.41T + 83T^{2} \)
89 \( 1 + (-4.61 - 8.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.37iT - 97T^{2} \)
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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

−9.196205100213994328225861040595, −8.791839548915748015811304707421, −7.35696405006091822307146526460, −6.94951964812842316110354649858, −6.18834747782618320765404886977, −5.36115495682610594107002878502, −4.49104098608967045517462545128, −3.06841774073620094552148286120, −2.76129620122557747056181677909, −1.39201848176935438789199603399, 0.69120349555966597590950749999, 1.59720021689794423390536203288, 3.18586713365662619391093494684, 3.78489040033913843383564370153, 4.95324896066119694168224639637, 5.63554336697605424573058798280, 6.35358667515558568456892387232, 7.45970528607448614392590477695, 7.930905652484975841005343567500, 8.935906764734872631285586413515

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