↑

Properties

Label 2-2268-21.17-c1-0-13
Degree $2$
Conductor $2268$
Sign $0.697 - 0.716i$
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  + (0.440 + 0.762i)5-s + (0.391 + 2.61i)7-s + (4.83 + 2.79i)11-s − 2.59i·13-s + (2.35 − 4.08i)17-s + (0.537 − 0.310i)19-s + (−3.15 + 1.82i)23-s + (2.11 − 3.65i)25-s − 5.33i·29-s + (9.56 + 5.51i)31-s + (−1.82 + 1.44i)35-s + (4.28 + 7.41i)37-s − 6.82·41-s + 3.65·43-s + (−3.88 − 6.73i)47-s + ⋯
L(s)  = 1  + (0.196 + 0.340i)5-s + (0.147 + 0.989i)7-s + (1.45 + 0.841i)11-s − 0.719i·13-s + (0.571 − 0.990i)17-s + (0.123 − 0.0712i)19-s + (−0.658 + 0.380i)23-s + (0.422 − 0.731i)25-s − 0.991i·29-s + (1.71 + 0.991i)31-s + (−0.308 + 0.245i)35-s + (0.703 + 1.21i)37-s − 1.06·41-s + 0.557·43-s + (−0.566 − 0.982i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.697 - 0.716i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.697 - 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.136732342\)
\(L(\frac12)\) \(\approx\) \(2.136732342\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.391 - 2.61i)T \)
good5 \( 1 + (-0.440 - 0.762i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.83 - 2.79i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.59iT - 13T^{2} \)
17 \( 1 + (-2.35 + 4.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.537 + 0.310i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.15 - 1.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.33iT - 29T^{2} \)
31 \( 1 + (-9.56 - 5.51i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.28 - 7.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 - 3.65T + 43T^{2} \)
47 \( 1 + (3.88 + 6.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.87 - 3.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.35 - 9.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.77 - 1.02i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.52 - 7.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.3iT - 71T^{2} \)
73 \( 1 + (3.43 + 1.98i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.01 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.43T + 83T^{2} \)
89 \( 1 + (-3.40 - 5.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.2iT - 97T^{2} \)
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

−9.152834867377482916746922597980, −8.397471912597402329557474819770, −7.60894318678623202878171432260, −6.63748816621305878033712412937, −6.13685965261042592697992538940, −5.13211892056006394861241422201, −4.38476374952433095779689912019, −3.17950124038955118637772012164, −2.40042852201235007280687109716, −1.18057323948061886956574720879, 0.907960498173570098587119527203, 1.74304498533577112762605676850, 3.35512798038106671432078649753, 4.01959640321523050660937983539, 4.77153347110834542797673400838, 6.01968718738441388075222459263, 6.45208511588654676759236029187, 7.39345477328158455547042476338, 8.219562344550563962353642318225, 8.924905400654074853213422405059

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