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Properties

Label 2-2268-63.16-c1-0-26
Degree $2$
Conductor $2268$
Sign $-0.214 + 0.976i$
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  + 1.03·5-s + (1.07 + 2.41i)7-s − 1.58·11-s + (−2.52 − 4.37i)13-s + (−2.58 − 4.47i)17-s + (−0.392 + 0.680i)19-s − 5.86·23-s − 3.93·25-s + (4.44 − 7.69i)29-s + (0.575 − 0.996i)31-s + (1.10 + 2.49i)35-s + (4.07 − 7.06i)37-s + (−3.87 − 6.70i)41-s + (1.26 − 2.19i)43-s + (−4.24 − 7.35i)47-s + ⋯
L(s)  = 1  + 0.461·5-s + (0.406 + 0.913i)7-s − 0.478·11-s + (−0.700 − 1.21i)13-s + (−0.626 − 1.08i)17-s + (−0.0901 + 0.156i)19-s − 1.22·23-s − 0.787·25-s + (0.825 − 1.42i)29-s + (0.103 − 0.179i)31-s + (0.187 + 0.421i)35-s + (0.670 − 1.16i)37-s + (−0.604 − 1.04i)41-s + (0.193 − 0.334i)43-s + (−0.619 − 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.214 + 0.976i$
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: \((2,\ 2268,\ (\ :1/2),\ -0.214 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.114774033\)
\(L(\frac12)\) \(\approx\) \(1.114774033\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.07 - 2.41i)T \)
good5 \( 1 - 1.03T + 5T^{2} \)
11 \( 1 + 1.58T + 11T^{2} \)
13 \( 1 + (2.52 + 4.37i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.58 + 4.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.392 - 0.680i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.86T + 23T^{2} \)
29 \( 1 + (-4.44 + 7.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.575 + 0.996i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.07 + 7.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.87 + 6.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.26 + 2.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.24 + 7.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.41 - 4.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.93 - 3.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.82 - 8.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.837 + 1.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 + (-3.04 - 5.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.15 - 7.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.12 + 12.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.69 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.67 - 4.63i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.761706666192673492011973136927, −8.018749612846578222324452720084, −7.44607935829195695711973738733, −6.27821586412644776334203622392, −5.58114292417273048179007346747, −5.03439118063053629714279591747, −3.94727506117385583571141894819, −2.54165243900902740189645437324, −2.22926191744209476519025877888, −0.35886743517351276645601851471, 1.46107567666792661649050290385, 2.28940109952868211874987040623, 3.62348142183726107718253868316, 4.50052836778015979704343453262, 5.09550694393416918582523481989, 6.44138162934173970926320511911, 6.66400183266040526387830507267, 7.898348089469415978506940438054, 8.247852391141230818083219528840, 9.420389147654390836619030827475

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