Properties

Label 2-12-4.3-c18-0-7
Degree $2$
Conductor $12$
Sign $0.912 + 0.408i$
Analytic cond. $24.6463$
Root an. cond. $4.96450$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−429. + 278. i)2-s − 1.13e4i·3-s + (1.07e5 − 2.39e5i)4-s − 3.52e6·5-s + (3.16e6 + 4.88e6i)6-s + 3.84e7i·7-s + (2.06e7 + 1.32e8i)8-s − 1.29e8·9-s + (1.51e9 − 9.82e8i)10-s + 2.54e9i·11-s + (−2.71e9 − 1.21e9i)12-s − 1.77e10·13-s + (−1.06e10 − 1.65e10i)14-s + 4.00e10i·15-s + (−4.58e10 − 5.12e10i)16-s + 3.74e10·17-s + ⋯
L(s)  = 1  + (−0.839 + 0.543i)2-s − 0.577i·3-s + (0.408 − 0.912i)4-s − 1.80·5-s + (0.314 + 0.484i)6-s + 0.951i·7-s + (0.154 + 0.988i)8-s − 0.333·9-s + (1.51 − 0.982i)10-s + 1.07i·11-s + (−0.527 − 0.235i)12-s − 1.67·13-s + (−0.517 − 0.798i)14-s + 1.04i·15-s + (−0.666 − 0.745i)16-s + 0.315·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.912 + 0.408i$
Analytic conductor: \(24.6463\)
Root analytic conductor: \(4.96450\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :9),\ 0.912 + 0.408i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.4352641376\)
\(L(\frac12)\) \(\approx\) \(0.4352641376\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (429. - 278. i)T \)
3 \( 1 + 1.13e4iT \)
good5 \( 1 + 3.52e6T + 3.81e12T^{2} \)
7 \( 1 - 3.84e7iT - 1.62e15T^{2} \)
11 \( 1 - 2.54e9iT - 5.55e18T^{2} \)
13 \( 1 + 1.77e10T + 1.12e20T^{2} \)
17 \( 1 - 3.74e10T + 1.40e22T^{2} \)
19 \( 1 + 4.75e11iT - 1.04e23T^{2} \)
23 \( 1 - 1.22e12iT - 3.24e24T^{2} \)
29 \( 1 + 8.73e12T + 2.10e26T^{2} \)
31 \( 1 + 1.96e13iT - 6.99e26T^{2} \)
37 \( 1 - 1.45e14T + 1.68e28T^{2} \)
41 \( 1 + 1.72e14T + 1.07e29T^{2} \)
43 \( 1 + 2.96e14iT - 2.52e29T^{2} \)
47 \( 1 - 2.69e14iT - 1.25e30T^{2} \)
53 \( 1 + 5.19e14T + 1.08e31T^{2} \)
59 \( 1 - 6.45e14iT - 7.50e31T^{2} \)
61 \( 1 - 3.10e15T + 1.36e32T^{2} \)
67 \( 1 + 3.87e16iT - 7.40e32T^{2} \)
71 \( 1 + 4.13e16iT - 2.10e33T^{2} \)
73 \( 1 - 4.63e16T + 3.46e33T^{2} \)
79 \( 1 - 1.75e17iT - 1.43e34T^{2} \)
83 \( 1 - 4.37e16iT - 3.49e34T^{2} \)
89 \( 1 + 9.76e16T + 1.22e35T^{2} \)
97 \( 1 + 5.14e17T + 5.77e35T^{2} \)
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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

−15.41339430377196946356893954899, −14.95553718391334038106028148843, −12.34161541380026035708116388448, −11.43440283134002688700978959685, −9.346012905685233418656173795363, −7.83368531720621209983306041989, −7.09232396605464701630446885170, −4.92338247409224915691833191321, −2.41098311443796937320340495980, −0.35599756356285737306153122415, 0.62806465097935001158389181037, 3.21617149273350670897417601184, 4.24235465867724776669794026119, 7.35613133604403223626615840206, 8.311268350433390301959059847583, 10.13608267949179571806935624918, 11.27784308877552964323246734648, 12.35660095793707763533337524085, 14.67479208604173224258633654472, 16.25252150787800626568958809988

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