↑

Properties

Label 2-2268-63.16-c1-0-12
Degree $2$
Conductor $2268$
Sign $0.975 - 0.220i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.16·5-s + (2 + 1.73i)7-s − 6.32·11-s + (−1.58 − 2.73i)17-s + (3.5 − 6.06i)19-s + 3.16·23-s + 5.00·25-s + (1.58 − 2.73i)29-s + (−1.5 + 2.59i)31-s + (−6.32 − 5.47i)35-s + (2 − 3.46i)37-s + (4.74 + 8.21i)41-s + (−2.5 + 4.33i)43-s + (4.74 + 8.21i)47-s + (1.00 + 6.92i)49-s + ⋯
L(s)  = 1  − 1.41·5-s + (0.755 + 0.654i)7-s − 1.90·11-s + (−0.383 − 0.664i)17-s + (0.802 − 1.39i)19-s + 0.659·23-s + 1.00·25-s + (0.293 − 0.508i)29-s + (−0.269 + 0.466i)31-s + (−1.06 − 0.925i)35-s + (0.328 − 0.569i)37-s + (0.740 + 1.28i)41-s + (−0.381 + 0.660i)43-s + (0.691 + 1.19i)47-s + (0.142 + 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.975 - 0.220i$
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: \((2,\ 2268,\ (\ :1/2),\ 0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.099789717\)
\(L(\frac12)\) \(\approx\) \(1.099789717\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 + 3.16T + 5T^{2} \)
11 \( 1 + 6.32T + 11T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.58 + 2.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.16T + 23T^{2} \)
29 \( 1 + (-1.58 + 2.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.74 - 8.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.74 - 8.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.74 - 8.21i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.32 + 10.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.16 + 5.47i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.74 - 8.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.878881568054151945285699765153, −8.135415326988957778445567812612, −7.62263493301341788049229487336, −7.05478605714377526284808287545, −5.70444950146417962431354088647, −4.89822464012421767253703353747, −4.46981679914674845531551966966, −3.02762324538868913848914381860, −2.53434934434147814591895155631, −0.68965095114519755446956732667, 0.63830872377694276892959890316, 2.14922218074812560373109029108, 3.42146584452410823947183659146, 4.03960659843102050734638198724, 4.99947911295292348153478464319, 5.61237190613386970370411008740, 7.05783999090275954170411028413, 7.54239033466336510167096721357, 8.153718150284784794498646038871, 8.593387969760950106633315042855

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