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Properties

Label 2-2268-63.20-c1-0-8
Degree $2$
Conductor $2268$
Sign $-0.873 - 0.486i$
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  + (1.5 + 2.59i)5-s + (0.5 + 2.59i)7-s + (−4.5 − 2.59i)11-s + (−3 + 1.73i)13-s + 6·17-s − 1.73i·19-s + (−4.5 + 2.59i)23-s + (−2 + 3.46i)25-s + (9 + 5.19i)29-s + (−4.5 + 2.59i)31-s + (−6 + 5.19i)35-s + 37-s + (−1.5 − 2.59i)41-s + (−5 + 8.66i)43-s + (−3 + 5.19i)47-s + ⋯
L(s)  = 1  + (0.670 + 1.16i)5-s + (0.188 + 0.981i)7-s + (−1.35 − 0.783i)11-s + (−0.832 + 0.480i)13-s + 1.45·17-s − 0.397i·19-s + (−0.938 + 0.541i)23-s + (−0.400 + 0.692i)25-s + (1.67 + 0.964i)29-s + (−0.808 + 0.466i)31-s + (−1.01 + 0.878i)35-s + 0.164·37-s + (−0.234 − 0.405i)41-s + (−0.762 + 1.32i)43-s + (−0.437 + 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.873 - 0.486i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.873 - 0.486i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.241958986\)
\(L(\frac12)\) \(\approx\) \(1.241958986\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.5 + 2.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 + (4.5 - 2.59i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-9 - 5.19i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.5 - 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5 - 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12 + 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.19iT - 71T^{2} \)
73 \( 1 - 3.46iT - 73T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3 + 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (6 + 3.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.528603442246488185769251734189, −8.477890694291114219936246941255, −7.83183766628998824800506525076, −7.00423361669982307031468396996, −6.10985067621884162647043114856, −5.52857665224749645380087159067, −4.79567275951271402140185945471, −3.06282524784053394291760056659, −2.88768169706976723533806931641, −1.74010894474790852585693735195, 0.40003101924768251832120590587, 1.60805998744677780583108325499, 2.65329439272408476644445287564, 3.95213466285539173409362591479, 4.92960570054791929779174645553, 5.25267778488987901645138101244, 6.24133944816750565531342282944, 7.46023629192446959198221374310, 7.85377480890382460872734237911, 8.564139481254832215585719344567

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