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Properties

Label 2-2268-63.4-c1-0-9
Degree $2$
Conductor $2268$
Sign $-0.0859 - 0.996i$
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  + 0.866·5-s + (−0.0665 + 2.64i)7-s + 3.51·11-s + (0.933 − 1.61i)13-s + (−3.25 + 5.64i)17-s + (2.69 + 4.66i)19-s − 8.64·23-s − 4.24·25-s + (1.75 + 3.04i)29-s + (0.933 + 1.61i)31-s + (−0.0576 + 2.29i)35-s + (1.39 + 2.40i)37-s + (5.19 − 8.99i)41-s + (2.89 + 5.00i)43-s + (3.08 − 5.33i)47-s + ⋯
L(s)  = 1  + 0.387·5-s + (−0.0251 + 0.999i)7-s + 1.05·11-s + (0.258 − 0.448i)13-s + (−0.790 + 1.36i)17-s + (0.617 + 1.06i)19-s − 1.80·23-s − 0.849·25-s + (0.326 + 0.565i)29-s + (0.167 + 0.290i)31-s + (−0.00975 + 0.387i)35-s + (0.228 + 0.395i)37-s + (0.810 − 1.40i)41-s + (0.440 + 0.763i)43-s + (0.449 − 0.778i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0859 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0859 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.0859 - 0.996i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.0859 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.653351111\)
\(L(\frac12)\) \(\approx\) \(1.653351111\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0665 - 2.64i)T \)
good5 \( 1 - 0.866T + 5T^{2} \)
11 \( 1 - 3.51T + 11T^{2} \)
13 \( 1 + (-0.933 + 1.61i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.25 - 5.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.69 - 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 8.64T + 23T^{2} \)
29 \( 1 + (-1.75 - 3.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.933 - 1.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.39 - 2.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.19 + 8.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.89 - 5.00i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.08 + 5.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.80 + 4.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.82 + 4.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.14 - 8.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.676 - 1.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.08T + 71T^{2} \)
73 \( 1 + (3.62 - 6.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.83 - 10.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.43 - 5.94i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.28 + 5.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.64 - 2.85i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.180016496476372354421148199623, −8.471262242339740184827990591462, −7.903817060181929576165868552038, −6.70125175371106600404407516032, −5.91338563039042403474496517300, −5.65833238228640242433581502555, −4.23619521904418674477200688349, −3.59637085067597675922138065736, −2.28651837820200821255288982225, −1.50020763609166861987319246775, 0.56644664193257713070160745640, 1.81365079270700213859782725209, 2.94242482064933308235438342460, 4.19985500921473634525173354624, 4.48400243343827945162501766074, 5.85980110438981897592816970694, 6.48301692505762134391155135881, 7.27036736007076967278986902543, 7.896079593005782523535783197907, 9.111316986708714566911851409676

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