↑

Properties

Label 2-12-3.2-c8-0-0
Degree $2$
Conductor $12$
Sign $-0.629 - 0.776i$
Analytic cond. $4.88854$
Root an. cond. $2.21100$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−51 − 62.9i)3-s + 1.13e3i·5-s − 3.09e3·7-s + (−1.35e3 + 6.41e3i)9-s + 1.13e3i·11-s − 7.29e3·13-s + (7.12e4 − 5.77e4i)15-s − 5.89e4i·17-s − 8.03e4·19-s + (1.57e5 + 1.94e5i)21-s + 9.74e4i·23-s − 8.92e5·25-s + (4.73e5 − 2.41e5i)27-s + 8.64e5i·29-s + 4.35e5·31-s + ⋯
L(s)  = 1  + (−0.629 − 0.776i)3-s + 1.81i·5-s − 1.28·7-s + (−0.207 + 0.978i)9-s + 0.0773i·11-s − 0.255·13-s + (1.40 − 1.14i)15-s − 0.705i·17-s − 0.616·19-s + (0.811 + 1.00i)21-s + 0.348i·23-s − 2.28·25-s + (0.890 − 0.455i)27-s + 1.22i·29-s + 0.472·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.776i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.629 - 0.776i$
Analytic conductor: \(4.88854\)
Root analytic conductor: \(2.21100\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :4),\ -0.629 - 0.776i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.229603 + 0.481621i\)
\(L(\frac12)\) \(\approx\) \(0.229603 + 0.481621i\)
\(L(5)\) 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 \( 1 + (51 + 62.9i)T \)
good5 \( 1 - 1.13e3iT - 3.90e5T^{2} \)
7 \( 1 + 3.09e3T + 5.76e6T^{2} \)
11 \( 1 - 1.13e3iT - 2.14e8T^{2} \)
13 \( 1 + 7.29e3T + 8.15e8T^{2} \)
17 \( 1 + 5.89e4iT - 6.97e9T^{2} \)
19 \( 1 + 8.03e4T + 1.69e10T^{2} \)
23 \( 1 - 9.74e4iT - 7.83e10T^{2} \)
29 \( 1 - 8.64e5iT - 5.00e11T^{2} \)
31 \( 1 - 4.35e5T + 8.52e11T^{2} \)
37 \( 1 - 1.15e6T + 3.51e12T^{2} \)
41 \( 1 - 2.71e6iT - 7.98e12T^{2} \)
43 \( 1 - 9.90e5T + 1.16e13T^{2} \)
47 \( 1 - 6.70e6iT - 2.38e13T^{2} \)
53 \( 1 + 1.00e7iT - 6.22e13T^{2} \)
59 \( 1 + 1.59e6iT - 1.46e14T^{2} \)
61 \( 1 - 1.93e7T + 1.91e14T^{2} \)
67 \( 1 + 2.80e7T + 4.06e14T^{2} \)
71 \( 1 - 3.36e7iT - 6.45e14T^{2} \)
73 \( 1 + 2.52e7T + 8.06e14T^{2} \)
79 \( 1 + 6.34e7T + 1.51e15T^{2} \)
83 \( 1 + 4.75e7iT - 2.25e15T^{2} \)
89 \( 1 - 7.82e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.95e7T + 7.83e15T^{2} \)
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

−18.76762627476353375259606061739, −17.71891058070302307748402858135, −16.07983255575924744869922157749, −14.44390812927988331574038749502, −13.01539244434248449812613017613, −11.40129985347850220984691901489, −10.09502317702145436350847680453, −7.22031344423970737463721534734, −6.27747016969976537408119544359, −2.83317309155976541842739968856, 0.34886356695735904396350483660, 4.25455643394907026497782402673, 5.91441747836610582575367729922, 8.820442190361542846253672113124, 10.01373990657277874261341800955, 12.10179778674517631898903364219, 13.09158384914558410691953819291, 15.49936661717884394921006257882, 16.50815603606336456160906881401, 17.21482746106720379639862545719

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