↑

Properties

Label 2-6-3.2-c28-0-5
Degree $2$
Conductor $6$
Sign $0.971 - 0.235i$
Analytic cond. $29.8010$
Root an. cond. $5.45903$
Motivic weight $28$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.15e4i·2-s + (−1.12e6 − 4.64e6i)3-s − 1.34e8·4-s + 2.00e9i·5-s + (5.38e10 − 1.30e10i)6-s − 2.77e11·7-s − 1.55e12i·8-s + (−2.03e13 + 1.04e13i)9-s − 2.32e13·10-s + 3.16e14i·11-s + (1.51e14 + 6.23e14i)12-s − 1.79e15·13-s − 3.21e15i·14-s + (9.32e15 − 2.25e15i)15-s + 1.80e16·16-s − 2.77e17i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.235 − 0.971i)3-s − 0.500·4-s + 0.328i·5-s + (0.687 − 0.166i)6-s − 0.409·7-s − 0.353i·8-s + (−0.889 + 0.457i)9-s − 0.232·10-s + 0.834i·11-s + (0.117 + 0.485i)12-s − 0.454·13-s − 0.289i·14-s + (0.319 − 0.0773i)15-s + 0.250·16-s − 1.65i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(29-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+14) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.971 - 0.235i$
Analytic conductor: \(29.8010\)
Root analytic conductor: \(5.45903\)
Motivic weight: \(28\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :14),\ 0.971 - 0.235i)\)

Particular Values

\(L(\frac{29}{2})\) \(\approx\) \(1.430097090\)
\(L(\frac12)\) \(\approx\) \(1.430097090\)
\(L(15)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.15e4iT \)
3 \( 1 + (1.12e6 + 4.64e6i)T \)
good5 \( 1 - 2.00e9iT - 3.72e19T^{2} \)
7 \( 1 + 2.77e11T + 4.59e23T^{2} \)
11 \( 1 - 3.16e14iT - 1.44e29T^{2} \)
13 \( 1 + 1.79e15T + 1.55e31T^{2} \)
17 \( 1 + 2.77e17iT - 2.83e34T^{2} \)
19 \( 1 - 7.80e17T + 6.38e35T^{2} \)
23 \( 1 - 6.21e18iT - 1.34e38T^{2} \)
29 \( 1 + 1.10e19iT - 8.85e40T^{2} \)
31 \( 1 - 5.88e20T + 5.72e41T^{2} \)
37 \( 1 - 1.39e22T + 8.12e43T^{2} \)
41 \( 1 - 3.87e22iT - 1.43e45T^{2} \)
43 \( 1 - 8.59e22T + 5.45e45T^{2} \)
47 \( 1 + 3.57e23iT - 6.58e46T^{2} \)
53 \( 1 + 8.74e22iT - 1.90e48T^{2} \)
59 \( 1 + 2.02e24iT - 3.83e49T^{2} \)
61 \( 1 - 1.78e25T + 9.75e49T^{2} \)
67 \( 1 - 2.42e25T + 1.34e51T^{2} \)
71 \( 1 + 8.89e25iT - 6.84e51T^{2} \)
73 \( 1 + 8.95e25T + 1.48e52T^{2} \)
79 \( 1 + 4.65e26T + 1.36e53T^{2} \)
83 \( 1 + 2.00e26iT - 5.42e53T^{2} \)
89 \( 1 - 2.85e27iT - 3.82e54T^{2} \)
97 \( 1 - 1.17e28T + 4.26e55T^{2} \)
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

−16.19265337079400233874020806504, −14.49660703379269581253256824549, −13.20107604188916198254030900458, −11.75118590247371027748458506247, −9.577135048099695085301640097263, −7.62330452740696735090380792948, −6.70302688003757548527104437503, −5.09393688793036887157753778239, −2.71828981751988883855022735580, −0.75883768888326410534631469847, 0.77216940605862822107311195148, 2.94832413268521414323534808234, 4.28426106428404526268425531708, 5.82900370521050567487972898631, 8.569311391955568673940921403420, 9.911468401453764121613462886924, 11.15081009886027874470649222199, 12.67401243921017400762169729041, 14.44958242154999881632058523907, 16.10227024313819759842402460369

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