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Properties

Label 2-2268-63.16-c1-0-9
Degree $2$
Conductor $2268$
Sign $-0.160 - 0.987i$
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  + 2.42·5-s + (0.710 + 2.54i)7-s − 4.70·11-s + (1.71 + 2.96i)13-s + (0.851 + 1.47i)17-s + (−0.641 + 1.11i)19-s + 1.12·23-s + 0.861·25-s + (−2.35 + 4.07i)29-s + (1.71 − 2.96i)31-s + (1.72 + 6.17i)35-s + (−4.27 + 7.40i)37-s + (1.85 + 3.21i)41-s + (−2.77 + 4.80i)43-s + (−5.91 − 10.2i)47-s + ⋯
L(s)  = 1  + 1.08·5-s + (0.268 + 0.963i)7-s − 1.41·11-s + (0.474 + 0.821i)13-s + (0.206 + 0.357i)17-s + (−0.147 + 0.254i)19-s + 0.234·23-s + 0.172·25-s + (−0.436 + 0.756i)29-s + (0.307 − 0.532i)31-s + (0.290 + 1.04i)35-s + (−0.702 + 1.21i)37-s + (0.290 + 0.502i)41-s + (−0.422 + 0.732i)43-s + (−0.862 − 1.49i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.160 - 0.987i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.160 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.731336236\)
\(L(\frac12)\) \(\approx\) \(1.731336236\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.710 - 2.54i)T \)
good5 \( 1 - 2.42T + 5T^{2} \)
11 \( 1 + 4.70T + 11T^{2} \)
13 \( 1 + (-1.71 - 2.96i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.851 - 1.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.641 - 1.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.12T + 23T^{2} \)
29 \( 1 + (2.35 - 4.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.71 + 2.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.27 - 7.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.85 - 3.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.77 - 4.80i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.91 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.13 - 8.88i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.06 + 3.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.62 - 8.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.56 + 9.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 + (1.06 + 1.85i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.26 - 12.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.21 + 7.29i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.04 + 13.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.12 - 14.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.206489548721169758042805767932, −8.522553383391670258079733903695, −7.86098744761838504451925753545, −6.74828400661651925694410864469, −5.97789269348909436108653677855, −5.39674534490728219714360479834, −4.66551805215900006129743256043, −3.28651693260839478317117687738, −2.32838408177465230684447699241, −1.60328604644358116945927535363, 0.55934053428233781162252547679, 1.89318752447547470278514728944, 2.84942289485843443095687514348, 3.88213154673551061039267051827, 5.07749453900304426468190849621, 5.49971143905617084856615747622, 6.43530830998232782541566605038, 7.37045914313953812415309418917, 7.948288078507669519550112250639, 8.796692896736426266494557233050

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