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Properties

Label 2-12-12.11-c19-0-2
Degree $2$
Conductor $12$
Sign $-0.710 - 0.703i$
Analytic cond. $27.4580$
Root an. cond. $5.24003$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−627. + 362. i)2-s + (8.68e3 − 3.29e4i)3-s + (2.61e5 − 4.54e5i)4-s + 6.03e6i·5-s + (6.49e6 + 2.38e7i)6-s − 9.28e7i·7-s + (2.07e5 + 3.79e8i)8-s + (−1.01e9 − 5.72e8i)9-s + (−2.18e9 − 3.78e9i)10-s − 6.13e9·11-s + (−1.26e10 − 1.25e10i)12-s + 4.99e10·13-s + (3.36e10 + 5.82e10i)14-s + (1.98e11 + 5.23e10i)15-s + (−1.37e11 − 2.37e11i)16-s − 2.01e11i·17-s + ⋯
L(s)  = 1  + (−0.865 + 0.500i)2-s + (0.254 − 0.967i)3-s + (0.499 − 0.866i)4-s + 1.38i·5-s + (0.263 + 0.964i)6-s − 0.870i·7-s + (0.000545 + 0.999i)8-s + (−0.870 − 0.492i)9-s + (−0.690 − 1.19i)10-s − 0.784·11-s + (−0.710 − 0.703i)12-s + 1.30·13-s + (0.435 + 0.753i)14-s + (1.33 + 0.351i)15-s + (−0.500 − 0.865i)16-s − 0.412i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.710 - 0.703i$
Analytic conductor: \(27.4580\)
Root analytic conductor: \(5.24003\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :19/2),\ -0.710 - 0.703i)\)

Particular Values

\(L(10)\) \(\approx\) \(0.4408918534\)
\(L(\frac12)\) \(\approx\) \(0.4408918534\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (627. - 362. i)T \)
3 \( 1 + (-8.68e3 + 3.29e4i)T \)
good5 \( 1 - 6.03e6iT - 1.90e13T^{2} \)
7 \( 1 + 9.28e7iT - 1.13e16T^{2} \)
11 \( 1 + 6.13e9T + 6.11e19T^{2} \)
13 \( 1 - 4.99e10T + 1.46e21T^{2} \)
17 \( 1 + 2.01e11iT - 2.39e23T^{2} \)
19 \( 1 - 4.67e11iT - 1.97e24T^{2} \)
23 \( 1 + 8.46e12T + 7.46e25T^{2} \)
29 \( 1 + 2.39e13iT - 6.10e27T^{2} \)
31 \( 1 - 1.41e14iT - 2.16e28T^{2} \)
37 \( 1 + 1.37e15T + 6.24e29T^{2} \)
41 \( 1 - 3.22e15iT - 4.39e30T^{2} \)
43 \( 1 - 5.84e15iT - 1.08e31T^{2} \)
47 \( 1 + 1.11e16T + 5.88e31T^{2} \)
53 \( 1 - 3.17e16iT - 5.77e32T^{2} \)
59 \( 1 - 1.59e16T + 4.42e33T^{2} \)
61 \( 1 - 1.40e17T + 8.34e33T^{2} \)
67 \( 1 + 3.36e16iT - 4.95e34T^{2} \)
71 \( 1 + 6.45e17T + 1.49e35T^{2} \)
73 \( 1 + 6.40e17T + 2.53e35T^{2} \)
79 \( 1 - 3.85e17iT - 1.13e36T^{2} \)
83 \( 1 - 5.19e17T + 2.90e36T^{2} \)
89 \( 1 + 5.87e17iT - 1.09e37T^{2} \)
97 \( 1 + 4.81e18T + 5.60e37T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.06506240917298621766821681217, −14.54628643433639213643352117754, −13.60249468522568144850710384178, −11.30770809203492987057578552821, −10.24061898430553254699949760550, −8.216365374303622088916151231846, −7.14797888397952129366237018358, −6.15626751508506223477278760567, −3.01236663856622698184619463465, −1.41837751060897160255949145847, 0.18298350708844538945477891312, 1.99871373631969360369405867854, 3.76684058797179403189858618887, 5.46702080236440612353464798824, 8.381912173768112592054986078548, 8.904249734862061996209587742207, 10.35209128250543943252318629240, 11.84115708533968029572619998777, 13.22186298208320411918120693699, 15.63135350527899007358146246252

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