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Properties

Label 2-2268-21.20-c3-0-52
Degree $2$
Conductor $2268$
Sign $0.595 - 0.803i$
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  + 12.0·5-s + (14.8 + 11.0i)7-s − 0.00255i·11-s − 7.00i·13-s + 28.3·17-s + 49.1i·19-s + 51.3i·23-s + 20.7·25-s − 113. i·29-s + 32.8i·31-s + (179. + 133. i)35-s − 101.·37-s − 22.4·41-s + 454.·43-s + 462.·47-s + ⋯
L(s)  = 1  + 1.07·5-s + (0.803 + 0.595i)7-s − 7.00e − 5i·11-s − 0.149i·13-s + 0.403·17-s + 0.594i·19-s + 0.465i·23-s + 0.165·25-s − 0.724i·29-s + 0.190i·31-s + (0.867 + 0.642i)35-s − 0.453·37-s − 0.0856·41-s + 1.61·43-s + 1.43·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.595 - 0.803i$
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: \((2,\ 2268,\ (\ :3/2),\ 0.595 - 0.803i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.206395918\)
\(L(\frac12)\) \(\approx\) \(3.206395918\)
\(L(\frac{5}{2})\) 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 \( 1 + (-14.8 - 11.0i)T \)
good5 \( 1 - 12.0T + 125T^{2} \)
11 \( 1 + 0.00255iT - 1.33e3T^{2} \)
13 \( 1 + 7.00iT - 2.19e3T^{2} \)
17 \( 1 - 28.3T + 4.91e3T^{2} \)
19 \( 1 - 49.1iT - 6.85e3T^{2} \)
23 \( 1 - 51.3iT - 1.21e4T^{2} \)
29 \( 1 + 113. iT - 2.43e4T^{2} \)
31 \( 1 - 32.8iT - 2.97e4T^{2} \)
37 \( 1 + 101.T + 5.06e4T^{2} \)
41 \( 1 + 22.4T + 6.89e4T^{2} \)
43 \( 1 - 454.T + 7.95e4T^{2} \)
47 \( 1 - 462.T + 1.03e5T^{2} \)
53 \( 1 - 567. iT - 1.48e5T^{2} \)
59 \( 1 + 291.T + 2.05e5T^{2} \)
61 \( 1 - 683. iT - 2.26e5T^{2} \)
67 \( 1 + 539.T + 3.00e5T^{2} \)
71 \( 1 + 307. iT - 3.57e5T^{2} \)
73 \( 1 + 495. iT - 3.89e5T^{2} \)
79 \( 1 - 649.T + 4.93e5T^{2} \)
83 \( 1 - 1.13e3T + 5.71e5T^{2} \)
89 \( 1 + 130.T + 7.04e5T^{2} \)
97 \( 1 - 1.45e3iT - 9.12e5T^{2} \)
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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

−9.014488171794675013482542065642, −7.939810295676275371884383462392, −7.41806009418803922166102063815, −6.08459915454019383453726668561, −5.79796156307180027579033477518, −4.96607428099348254754074607072, −3.97949246972395248267881941252, −2.73779504915975662742887763558, −1.96520999970496208804134347371, −1.07624521205883288199106791859, 0.66672279141015824974713703688, 1.67298776993555480457059678614, 2.48276752387012228249448372769, 3.69257173218362039065096313788, 4.67362382948786390781147663303, 5.36624222416245672780726906842, 6.17701788647602491829247289905, 7.02960881368673337414333006723, 7.73229898449634752221224707445, 8.644568270675773197172557902025

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