↑

Properties

Label 2-2268-21.20-c3-0-74
Degree $2$
Conductor $2268$
Sign $0.785 + 0.619i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 10.9·5-s + (11.4 − 14.5i)7-s + 15.9i·11-s − 89.4i·13-s + 106.·17-s + 8.31i·19-s + 143. i·23-s − 4.13·25-s + 149. i·29-s − 42.9i·31-s + (126. − 159. i)35-s + 390.·37-s + 344.·41-s − 57.1·43-s + 16.2·47-s + ⋯
L(s)  = 1  + 0.983·5-s + (0.619 − 0.785i)7-s + 0.437i·11-s − 1.90i·13-s + 1.52·17-s + 0.100i·19-s + 1.29i·23-s − 0.0330·25-s + 0.959i·29-s − 0.248i·31-s + (0.609 − 0.772i)35-s + 1.73·37-s + 1.31·41-s − 0.202·43-s + 0.0504·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.785 + 0.619i$
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: \((2,\ 2268,\ (\ :3/2),\ 0.785 + 0.619i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.433946460\)
\(L(\frac12)\) \(\approx\) \(3.433946460\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.4 + 14.5i)T \)
good5 \( 1 - 10.9T + 125T^{2} \)
11 \( 1 - 15.9iT - 1.33e3T^{2} \)
13 \( 1 + 89.4iT - 2.19e3T^{2} \)
17 \( 1 - 106.T + 4.91e3T^{2} \)
19 \( 1 - 8.31iT - 6.85e3T^{2} \)
23 \( 1 - 143. iT - 1.21e4T^{2} \)
29 \( 1 - 149. iT - 2.43e4T^{2} \)
31 \( 1 + 42.9iT - 2.97e4T^{2} \)
37 \( 1 - 390.T + 5.06e4T^{2} \)
41 \( 1 - 344.T + 6.89e4T^{2} \)
43 \( 1 + 57.1T + 7.95e4T^{2} \)
47 \( 1 - 16.2T + 1.03e5T^{2} \)
53 \( 1 - 445. iT - 1.48e5T^{2} \)
59 \( 1 - 386.T + 2.05e5T^{2} \)
61 \( 1 - 485. iT - 2.26e5T^{2} \)
67 \( 1 - 503.T + 3.00e5T^{2} \)
71 \( 1 + 751. iT - 3.57e5T^{2} \)
73 \( 1 + 507. iT - 3.89e5T^{2} \)
79 \( 1 + 763.T + 4.93e5T^{2} \)
83 \( 1 + 1.21e3T + 5.71e5T^{2} \)
89 \( 1 + 425.T + 7.04e5T^{2} \)
97 \( 1 + 571. iT - 9.12e5T^{2} \)
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

−8.482012375070566731182810023385, −7.57641985965640675032432148990, −7.42669973025376376452290914676, −5.85769678823208315691105159704, −5.66547256290313970651707733715, −4.70687675975875701955328573591, −3.59879448254563500279229973287, −2.74305027109566295561856158776, −1.50651108316001192642775420279, −0.812802776721728675298109259852, 0.986202898451648788877838722503, 2.01653103989798505764774261693, 2.62351526897207287310202049729, 4.00320783477738342099725389037, 4.81899102537292703286144972577, 5.78518394122498210898534757014, 6.18320758558715763372221327336, 7.14234311508037800512418058527, 8.180209809767602485888483865081, 8.722052816125631820100307252081

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