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Properties

Label 2-2268-21.20-c3-0-5
Degree $2$
Conductor $2268$
Sign $-0.989 + 0.147i$
Analytic cond. $133.816$
Root an. cond. $11.5679$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.3·5-s + (2.73 + 18.3i)7-s + 31.3i·11-s − 45.0i·13-s − 62.4·17-s + 132. i·19-s + 67.8i·23-s − 18.2·25-s − 134. i·29-s − 30.0i·31-s + (28.2 + 189. i)35-s + 40.4·37-s + 79.5·41-s − 322.·43-s − 342.·47-s + ⋯
L(s)  = 1  + 0.923·5-s + (0.147 + 0.989i)7-s + 0.858i·11-s − 0.960i·13-s − 0.891·17-s + 1.60i·19-s + 0.615i·23-s − 0.146·25-s − 0.862i·29-s − 0.173i·31-s + (0.136 + 0.913i)35-s + 0.179·37-s + 0.302·41-s − 1.14·43-s − 1.06·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.147i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.989 + 0.147i$
Analytic conductor: \(133.816\)
Root analytic conductor: \(11.5679\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :3/2),\ -0.989 + 0.147i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6525742518\)
\(L(\frac12)\) \(\approx\) \(0.6525742518\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.73 - 18.3i)T \)
good5 \( 1 - 10.3T + 125T^{2} \)
11 \( 1 - 31.3iT - 1.33e3T^{2} \)
13 \( 1 + 45.0iT - 2.19e3T^{2} \)
17 \( 1 + 62.4T + 4.91e3T^{2} \)
19 \( 1 - 132. iT - 6.85e3T^{2} \)
23 \( 1 - 67.8iT - 1.21e4T^{2} \)
29 \( 1 + 134. iT - 2.43e4T^{2} \)
31 \( 1 + 30.0iT - 2.97e4T^{2} \)
37 \( 1 - 40.4T + 5.06e4T^{2} \)
41 \( 1 - 79.5T + 6.89e4T^{2} \)
43 \( 1 + 322.T + 7.95e4T^{2} \)
47 \( 1 + 342.T + 1.03e5T^{2} \)
53 \( 1 - 64.9iT - 1.48e5T^{2} \)
59 \( 1 - 158.T + 2.05e5T^{2} \)
61 \( 1 + 570. iT - 2.26e5T^{2} \)
67 \( 1 + 301.T + 3.00e5T^{2} \)
71 \( 1 - 719. iT - 3.57e5T^{2} \)
73 \( 1 + 558. iT - 3.89e5T^{2} \)
79 \( 1 + 913.T + 4.93e5T^{2} \)
83 \( 1 + 704.T + 5.71e5T^{2} \)
89 \( 1 + 700.T + 7.04e5T^{2} \)
97 \( 1 - 234. iT - 9.12e5T^{2} \)
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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

−9.157302818231758444834084089882, −8.277052599028122913602803634993, −7.68949217089069534416543006058, −6.55289018075987981949992539415, −5.85881963306160551570448883127, −5.33546886264077577482446825201, −4.35570695261092567099494882861, −3.18879372929481468422497610740, −2.16623554757577041492865770893, −1.60065482579598482613137123309, 0.11675120133886511277020370077, 1.24764276631137009426507796551, 2.22895697825509658040372747498, 3.24707375629429476251146480868, 4.35966057054789321651423309849, 4.95521693532541535966560303072, 6.06017670254079617667325938229, 6.72949570639046447046700355472, 7.26692165718623476714021526608, 8.512743080487235798319484847167

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