↑

Properties

Label 2-12-3.2-c20-0-2
Degree $2$
Conductor $12$
Sign $0.332 - 0.943i$
Analytic cond. $30.4216$
Root an. cond. $5.51558$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.96e4 − 5.56e4i)3-s + 7.88e6i·5-s + 3.14e8·7-s + (−2.71e9 − 2.18e9i)9-s + 3.97e10i·11-s − 2.60e11·13-s + (4.39e11 + 1.54e11i)15-s − 4.22e10i·17-s + 1.21e12·19-s + (6.17e12 − 1.75e13i)21-s + 4.94e13i·23-s + 3.31e13·25-s + (−1.75e14 + 1.08e14i)27-s + 4.36e14i·29-s + 3.04e14·31-s + ⋯
L(s)  = 1  + (0.332 − 0.943i)3-s + 0.807i·5-s + 1.11·7-s + (−0.779 − 0.626i)9-s + 1.53i·11-s − 1.89·13-s + (0.761 + 0.268i)15-s − 0.0209i·17-s + 0.198·19-s + (0.369 − 1.05i)21-s + 1.19i·23-s + 0.348·25-s + (−0.850 + 0.526i)27-s + 1.03i·29-s + 0.371·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.332 - 0.943i$
Analytic conductor: \(30.4216\)
Root analytic conductor: \(5.51558\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :10),\ 0.332 - 0.943i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(1.722779750\)
\(L(\frac12)\) \(\approx\) \(1.722779750\)
\(L(11)\) 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 \( 1 + (-1.96e4 + 5.56e4i)T \)
good5 \( 1 - 7.88e6iT - 9.53e13T^{2} \)
7 \( 1 - 3.14e8T + 7.97e16T^{2} \)
11 \( 1 - 3.97e10iT - 6.72e20T^{2} \)
13 \( 1 + 2.60e11T + 1.90e22T^{2} \)
17 \( 1 + 4.22e10iT - 4.06e24T^{2} \)
19 \( 1 - 1.21e12T + 3.75e25T^{2} \)
23 \( 1 - 4.94e13iT - 1.71e27T^{2} \)
29 \( 1 - 4.36e14iT - 1.76e29T^{2} \)
31 \( 1 - 3.04e14T + 6.71e29T^{2} \)
37 \( 1 - 1.82e15T + 2.31e31T^{2} \)
41 \( 1 - 2.29e16iT - 1.80e32T^{2} \)
43 \( 1 - 1.37e16T + 4.67e32T^{2} \)
47 \( 1 - 4.50e16iT - 2.76e33T^{2} \)
53 \( 1 + 1.96e17iT - 3.05e34T^{2} \)
59 \( 1 + 4.53e17iT - 2.61e35T^{2} \)
61 \( 1 - 1.15e17T + 5.08e35T^{2} \)
67 \( 1 - 7.02e17T + 3.32e36T^{2} \)
71 \( 1 - 4.29e18iT - 1.05e37T^{2} \)
73 \( 1 + 7.58e18T + 1.84e37T^{2} \)
79 \( 1 - 1.37e19T + 8.96e37T^{2} \)
83 \( 1 + 9.70e18iT - 2.40e38T^{2} \)
89 \( 1 + 1.74e18iT - 9.72e38T^{2} \)
97 \( 1 + 1.87e19T + 5.43e39T^{2} \)
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

−14.87290547709257357733076857745, −14.44218380243875932675537164431, −12.64607897165756176686635733431, −11.49949194116428416731069646615, −9.713191817878722121195054797323, −7.74428668061471525170786985312, −7.00103070522047887866240174030, −4.91019485945021819773781449652, −2.68453808917971367830837421141, −1.58799641841920536786058420123, 0.50391776932807223405212972725, 2.52466835753088437537607539666, 4.39100282481864034419067828339, 5.37892058146146291869100120963, 7.999452846221085664639183192355, 9.041270322927227056675923991156, 10.60361280134484344984579225214, 11.98196663640804679868148603538, 13.89606034809589167956103912861, 14.91890692735435999708547704093

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