Properties

Label 2-12-4.3-c18-0-12
Degree $2$
Conductor $12$
Sign $-0.834 + 0.551i$
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  + (−450. − 242. i)2-s − 1.13e4i·3-s + (1.44e5 + 2.18e5i)4-s − 7.80e5·5-s + (−2.75e6 + 5.12e6i)6-s + 2.45e7i·7-s + (−1.20e7 − 1.33e8i)8-s − 1.29e8·9-s + (3.52e8 + 1.89e8i)10-s − 5.20e8i·11-s + (2.48e9 − 1.64e9i)12-s + 1.83e10·13-s + (5.95e9 − 1.10e10i)14-s + 8.87e9i·15-s + (−2.69e10 + 6.31e10i)16-s + 3.86e10·17-s + ⋯
L(s)  = 1  + (−0.880 − 0.473i)2-s − 0.577i·3-s + (0.551 + 0.834i)4-s − 0.399·5-s + (−0.273 + 0.508i)6-s + 0.608i·7-s + (−0.0899 − 0.995i)8-s − 0.333·9-s + (0.352 + 0.189i)10-s − 0.220i·11-s + (0.481 − 0.318i)12-s + 1.73·13-s + (0.288 − 0.535i)14-s + 0.230i·15-s + (−0.392 + 0.919i)16-s + 0.325·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.834 + 0.551i$
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: \((2,\ 12,\ (\ :9),\ -0.834 + 0.551i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.7535203289\)
\(L(\frac12)\) \(\approx\) \(0.7535203289\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (450. + 242. i)T \)
3 \( 1 + 1.13e4iT \)
good5 \( 1 + 7.80e5T + 3.81e12T^{2} \)
7 \( 1 - 2.45e7iT - 1.62e15T^{2} \)
11 \( 1 + 5.20e8iT - 5.55e18T^{2} \)
13 \( 1 - 1.83e10T + 1.12e20T^{2} \)
17 \( 1 - 3.86e10T + 1.40e22T^{2} \)
19 \( 1 + 8.98e10iT - 1.04e23T^{2} \)
23 \( 1 + 1.99e12iT - 3.24e24T^{2} \)
29 \( 1 + 1.93e13T + 2.10e26T^{2} \)
31 \( 1 + 1.86e13iT - 6.99e26T^{2} \)
37 \( 1 + 9.76e13T + 1.68e28T^{2} \)
41 \( 1 + 1.26e14T + 1.07e29T^{2} \)
43 \( 1 - 4.41e14iT - 2.52e29T^{2} \)
47 \( 1 + 1.38e15iT - 1.25e30T^{2} \)
53 \( 1 - 1.00e15T + 1.08e31T^{2} \)
59 \( 1 + 1.06e16iT - 7.50e31T^{2} \)
61 \( 1 + 2.00e16T + 1.36e32T^{2} \)
67 \( 1 + 5.21e16iT - 7.40e32T^{2} \)
71 \( 1 + 6.60e16iT - 2.10e33T^{2} \)
73 \( 1 + 7.78e14T + 3.46e33T^{2} \)
79 \( 1 - 5.19e15iT - 1.43e34T^{2} \)
83 \( 1 + 4.99e16iT - 3.49e34T^{2} \)
89 \( 1 - 2.80e17T + 1.22e35T^{2} \)
97 \( 1 - 3.77e17T + 5.77e35T^{2} \)
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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

−15.44722245667614977895288607948, −13.33789843429241489855775486225, −11.99371901122631762009477943321, −10.89766536626336160362523148838, −8.993168447944542059753626991013, −7.911111245391843001218365210950, −6.23813310620557043452688475327, −3.50155098380926206264942170601, −1.85073260369454570068620392414, −0.36754790074485132761018953840, 1.32095800435588858057878595481, 3.74474991282196542436688938599, 5.75708897925999851451328569737, 7.47724667161335808260004886327, 8.865787081427954403836392138784, 10.31520474836808229051743599641, 11.43133134312995181201538123848, 13.80544010641625929288334041911, 15.33827953115205132611273868682, 16.21595766036078595484722430102

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