↑

Properties

Label 2-2268-21.17-c1-0-30
Degree $2$
Conductor $2268$
Sign $-0.803 - 0.595i$
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  + (−1.43 − 2.48i)5-s + (−0.736 − 2.54i)7-s + (−2.34 − 1.35i)11-s − 3.68i·13-s + (−3.22 + 5.58i)17-s + (2.73 − 1.58i)19-s + (2.59 − 1.49i)23-s + (−1.61 + 2.79i)25-s − 2.86i·29-s + (−8.26 − 4.77i)31-s + (−5.25 + 5.47i)35-s + (−1.70 − 2.95i)37-s − 1.58·41-s + 9.35·43-s + (5.65 + 9.79i)47-s + ⋯
L(s)  = 1  + (−0.641 − 1.11i)5-s + (−0.278 − 0.960i)7-s + (−0.708 − 0.408i)11-s − 1.02i·13-s + (−0.781 + 1.35i)17-s + (0.628 − 0.362i)19-s + (0.540 − 0.311i)23-s + (−0.322 + 0.558i)25-s − 0.532i·29-s + (−1.48 − 0.857i)31-s + (−0.888 + 0.925i)35-s + (−0.280 − 0.485i)37-s − 0.248·41-s + 1.42·43-s + (0.824 + 1.42i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.803 - 0.595i$
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: \((2,\ 2268,\ (\ :1/2),\ -0.803 - 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4714835783\)
\(L(\frac12)\) \(\approx\) \(0.4714835783\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.736 + 2.54i)T \)
good5 \( 1 + (1.43 + 2.48i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.34 + 1.35i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.68iT - 13T^{2} \)
17 \( 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.73 + 1.58i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.59 + 1.49i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.86iT - 29T^{2} \)
31 \( 1 + (8.26 + 4.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.70 + 2.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.58T + 41T^{2} \)
43 \( 1 - 9.35T + 43T^{2} \)
47 \( 1 + (-5.65 - 9.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.16 + 1.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.33 - 7.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.566 + 0.327i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.86 - 6.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.86iT - 71T^{2} \)
73 \( 1 + (-11.0 - 6.39i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.59 + 4.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + (3.14 + 5.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.2iT - 97T^{2} \)
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

−8.526092788214803823107003980865, −7.76942790125419748433245833035, −7.30719758604209267683724550925, −6.06391401797298384911084724271, −5.37294316561956276494025400709, −4.36910427104120853364272984602, −3.84149999460771556426447304496, −2.70225009876622657872084113650, −1.13099043237602183190385246549, −0.17713055221114831003568051503, 1.98465682380980792489177961671, 2.87105441428086956788756484754, 3.57092369669543357317943990500, 4.79998536893559878744019510951, 5.48421089001603926094530159251, 6.63909052698721529034466515718, 7.11405823983201859914306607853, 7.73533703138819989997901321638, 8.950593919651972458373770958872, 9.273433115787250159043336985086

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