Properties

Label 2-12-4.3-c20-0-11
Degree $2$
Conductor $12$
Sign $0.548 - 0.836i$
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  + (981. + 293. i)2-s − 3.40e4i·3-s + (8.76e5 + 5.75e5i)4-s + 1.08e7·5-s + (9.99e6 − 3.34e7i)6-s + 4.06e8i·7-s + (6.91e8 + 8.21e8i)8-s − 1.16e9·9-s + (1.06e10 + 3.17e9i)10-s + 2.22e9i·11-s + (1.96e10 − 2.98e10i)12-s − 1.75e11·13-s + (−1.19e11 + 3.99e11i)14-s − 3.69e11i·15-s + (4.37e11 + 1.00e12i)16-s + 3.29e12·17-s + ⋯
L(s)  = 1  + (0.958 + 0.286i)2-s − 0.577i·3-s + (0.836 + 0.548i)4-s + 1.11·5-s + (0.165 − 0.553i)6-s + 1.44i·7-s + (0.644 + 0.764i)8-s − 0.333·9-s + (1.06 + 0.317i)10-s + 0.0857i·11-s + (0.316 − 0.482i)12-s − 1.27·13-s + (−0.412 + 1.38i)14-s − 0.641i·15-s + (0.398 + 0.917i)16-s + 1.63·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.548 - 0.836i$
Analytic conductor: \(30.4216\)
Root analytic conductor: \(5.51558\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :10),\ 0.548 - 0.836i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(4.396730859\)
\(L(\frac12)\) \(\approx\) \(4.396730859\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-981. - 293. i)T \)
3 \( 1 + 3.40e4iT \)
good5 \( 1 - 1.08e7T + 9.53e13T^{2} \)
7 \( 1 - 4.06e8iT - 7.97e16T^{2} \)
11 \( 1 - 2.22e9iT - 6.72e20T^{2} \)
13 \( 1 + 1.75e11T + 1.90e22T^{2} \)
17 \( 1 - 3.29e12T + 4.06e24T^{2} \)
19 \( 1 - 1.26e12iT - 3.75e25T^{2} \)
23 \( 1 - 3.97e13iT - 1.71e27T^{2} \)
29 \( 1 - 2.63e14T + 1.76e29T^{2} \)
31 \( 1 + 8.61e14iT - 6.71e29T^{2} \)
37 \( 1 + 1.00e15T + 2.31e31T^{2} \)
41 \( 1 - 2.19e16T + 1.80e32T^{2} \)
43 \( 1 - 3.71e16iT - 4.67e32T^{2} \)
47 \( 1 + 8.27e16iT - 2.76e33T^{2} \)
53 \( 1 - 1.21e17T + 3.05e34T^{2} \)
59 \( 1 + 5.44e17iT - 2.61e35T^{2} \)
61 \( 1 + 4.97e17T + 5.08e35T^{2} \)
67 \( 1 + 3.24e18iT - 3.32e36T^{2} \)
71 \( 1 + 2.30e18iT - 1.05e37T^{2} \)
73 \( 1 + 2.29e18T + 1.84e37T^{2} \)
79 \( 1 + 7.16e17iT - 8.96e37T^{2} \)
83 \( 1 + 5.66e18iT - 2.40e38T^{2} \)
89 \( 1 + 3.74e19T + 9.72e38T^{2} \)
97 \( 1 + 1.04e20T + 5.43e39T^{2} \)
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

−15.07236523235590467246581977039, −14.07649622815719514760681272116, −12.70257523936827382773025393274, −11.85502626483991062241106763896, −9.649328467441483175209956955985, −7.74704515095982781767802985374, −6.05325843700975319083000692114, −5.28528576782117221014198906811, −2.82163598485123000013418008313, −1.83553867055667822126213093198, 1.03605290575681769883656098818, 2.75017816929496599713304893402, 4.30026798414004743408414831398, 5.57915748123855025152086070614, 7.18815226382154521255242334094, 9.892217572629989890898071899914, 10.54001302744964590005113257312, 12.40273951434800383973091574662, 13.89823887714795798209967374938, 14.49147287080608916870006139245

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