↑

Properties

Label 2-12-12.11-c11-0-8
Degree $2$
Conductor $12$
Sign $0.962 + 0.269i$
Analytic cond. $9.22011$
Root an. cond. $3.03646$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−41.7 − 17.5i)2-s + (364. − 210. i)3-s + (1.43e3 + 1.46e3i)4-s − 505. i·5-s + (−1.88e4 + 2.38e3i)6-s + 7.75e4i·7-s + (−3.38e4 − 8.62e4i)8-s + (8.82e4 − 1.53e5i)9-s + (−8.88e3 + 2.10e4i)10-s + 7.31e5·11-s + (8.30e5 + 2.32e5i)12-s + 1.14e6·13-s + (1.36e6 − 3.23e6i)14-s + (−1.06e5 − 1.84e5i)15-s + (−1.03e5 + 4.19e6i)16-s − 3.04e6i·17-s + ⋯
L(s)  = 1  + (−0.921 − 0.388i)2-s + (0.865 − 0.500i)3-s + (0.698 + 0.715i)4-s − 0.0723i·5-s + (−0.992 + 0.125i)6-s + 1.74i·7-s + (−0.365 − 0.930i)8-s + (0.498 − 0.866i)9-s + (−0.0280 + 0.0666i)10-s + 1.36·11-s + (0.962 + 0.269i)12-s + 0.853·13-s + (0.677 − 1.60i)14-s + (−0.0362 − 0.0625i)15-s + (−0.0247 + 0.999i)16-s − 0.519i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.962 + 0.269i$
Analytic conductor: \(9.22011\)
Root analytic conductor: \(3.03646\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :11/2),\ 0.962 + 0.269i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.62462 - 0.223339i\)
\(L(\frac12)\) \(\approx\) \(1.62462 - 0.223339i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (41.7 + 17.5i)T \)
3 \( 1 + (-364. + 210. i)T \)
good5 \( 1 + 505. iT - 4.88e7T^{2} \)
7 \( 1 - 7.75e4iT - 1.97e9T^{2} \)
11 \( 1 - 7.31e5T + 2.85e11T^{2} \)
13 \( 1 - 1.14e6T + 1.79e12T^{2} \)
17 \( 1 + 3.04e6iT - 3.42e13T^{2} \)
19 \( 1 + 4.99e6iT - 1.16e14T^{2} \)
23 \( 1 - 5.14e6T + 9.52e14T^{2} \)
29 \( 1 - 1.44e8iT - 1.22e16T^{2} \)
31 \( 1 - 2.51e7iT - 2.54e16T^{2} \)
37 \( 1 + 1.69e8T + 1.77e17T^{2} \)
41 \( 1 + 3.25e8iT - 5.50e17T^{2} \)
43 \( 1 - 1.17e9iT - 9.29e17T^{2} \)
47 \( 1 + 1.82e9T + 2.47e18T^{2} \)
53 \( 1 - 5.85e7iT - 9.26e18T^{2} \)
59 \( 1 + 5.59e8T + 3.01e19T^{2} \)
61 \( 1 - 8.08e9T + 4.35e19T^{2} \)
67 \( 1 + 2.71e9iT - 1.22e20T^{2} \)
71 \( 1 + 1.89e10T + 2.31e20T^{2} \)
73 \( 1 - 1.88e10T + 3.13e20T^{2} \)
79 \( 1 + 4.26e10iT - 7.47e20T^{2} \)
83 \( 1 + 3.76e10T + 1.28e21T^{2} \)
89 \( 1 + 5.79e10iT - 2.77e21T^{2} \)
97 \( 1 + 9.56e10T + 7.15e21T^{2} \)
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

−17.92065539434707032255699574229, −16.03363762822770361016271814712, −14.71712542863246213359675681845, −12.71436040144732391250115822201, −11.60167956480674652430819381769, −9.222117319687313251871707789049, −8.618630848225472385813412251780, −6.63263909970939395996462747342, −3.04922691432557963364513434730, −1.47951982423957278193418649219, 1.25737558067352654367529404567, 3.89660881132630157398980338784, 6.83967918590782526722180832005, 8.319278371909944962969518598584, 9.812005193857193280010660107732, 10.95771354007402067094799586534, 13.74616179247402510354779224119, 14.78117250553054288150859195790, 16.35346870566908142315926056603, 17.22835133600602362966098933034

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