Properties

Label 2-12-4.3-c16-0-0
Degree $2$
Conductor $12$
Sign $-0.919 - 0.392i$
Analytic cond. $19.4789$
Root an. cond. $4.41349$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−141. − 213. i)2-s + 3.78e3i·3-s + (−2.56e4 + 6.02e4i)4-s + 1.10e5·5-s + (8.09e5 − 5.34e5i)6-s + 7.35e6i·7-s + (1.65e7 − 3.02e6i)8-s − 1.43e7·9-s + (−1.55e7 − 2.35e7i)10-s − 2.79e8i·11-s + (−2.28e8 − 9.73e7i)12-s + 2.85e7·13-s + (1.57e9 − 1.03e9i)14-s + 4.17e8i·15-s + (−2.97e9 − 3.09e9i)16-s − 6.15e9·17-s + ⋯
L(s)  = 1  + (−0.551 − 0.834i)2-s + 0.577i·3-s + (−0.392 + 0.919i)4-s + 0.281·5-s + (0.481 − 0.318i)6-s + 1.27i·7-s + (0.983 − 0.180i)8-s − 0.333·9-s + (−0.155 − 0.235i)10-s − 1.30i·11-s + (−0.531 − 0.226i)12-s + 0.0350·13-s + (1.06 − 0.703i)14-s + 0.162i·15-s + (−0.692 − 0.721i)16-s − 0.881·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.919 - 0.392i$
Analytic conductor: \(19.4789\)
Root analytic conductor: \(4.41349\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :8),\ -0.919 - 0.392i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.0522929 + 0.256088i\)
\(L(\frac12)\) \(\approx\) \(0.0522929 + 0.256088i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (141. + 213. i)T \)
3 \( 1 - 3.78e3iT \)
good5 \( 1 - 1.10e5T + 1.52e11T^{2} \)
7 \( 1 - 7.35e6iT - 3.32e13T^{2} \)
11 \( 1 + 2.79e8iT - 4.59e16T^{2} \)
13 \( 1 - 2.85e7T + 6.65e17T^{2} \)
17 \( 1 + 6.15e9T + 4.86e19T^{2} \)
19 \( 1 - 2.23e10iT - 2.88e20T^{2} \)
23 \( 1 - 1.78e10iT - 6.13e21T^{2} \)
29 \( 1 + 4.59e11T + 2.50e23T^{2} \)
31 \( 1 + 5.70e11iT - 7.27e23T^{2} \)
37 \( 1 + 6.55e12T + 1.23e25T^{2} \)
41 \( 1 - 6.32e12T + 6.37e25T^{2} \)
43 \( 1 + 1.18e13iT - 1.36e26T^{2} \)
47 \( 1 + 2.75e13iT - 5.66e26T^{2} \)
53 \( 1 + 1.04e14T + 3.87e27T^{2} \)
59 \( 1 - 1.51e14iT - 2.15e28T^{2} \)
61 \( 1 + 2.11e14T + 3.67e28T^{2} \)
67 \( 1 + 3.36e14iT - 1.64e29T^{2} \)
71 \( 1 - 2.76e14iT - 4.16e29T^{2} \)
73 \( 1 - 9.79e14T + 6.50e29T^{2} \)
79 \( 1 - 5.71e14iT - 2.30e30T^{2} \)
83 \( 1 - 3.91e15iT - 5.07e30T^{2} \)
89 \( 1 - 2.27e15T + 1.54e31T^{2} \)
97 \( 1 + 1.11e16T + 6.14e31T^{2} \)
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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

−16.89533769319960491621028441634, −15.60298074888930185005860643638, −13.75258501189003432755880797681, −12.11826612372035170035633862724, −10.91353426323243800644726200298, −9.402168865281929278808306140254, −8.369017977414930818132044059258, −5.66989325677024470593765192691, −3.58331158377942704938214291812, −2.04935157563990595325427594418, 0.11104524376596599128944118466, 1.68541957405731364780160598864, 4.64046886754389004867515509555, 6.65162500999674745979240990012, 7.56866197135839295960833601998, 9.379010056203904285171411085002, 10.82015235494292259588077855680, 13.09355428995520038384075876613, 14.18877754967959204713699225371, 15.66652823432815135262648610227

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