Properties

Label 2-12-4.3-c8-0-3
Degree $2$
Conductor $12$
Sign $0.890 - 0.455i$
Analytic cond. $4.88854$
Root an. cond. $2.21100$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (13.6 + 8.35i)2-s − 46.7i·3-s + (116. + 227. i)4-s + 904.·5-s + (390. − 638. i)6-s + 888. i·7-s + (−313. + 4.08e3i)8-s − 2.18e3·9-s + (1.23e4 + 7.55e3i)10-s − 1.44e4i·11-s + (1.06e4 − 5.44e3i)12-s − 1.16e4·13-s + (−7.41e3 + 1.21e4i)14-s − 4.22e4i·15-s + (−3.83e4 + 5.31e4i)16-s − 1.28e5·17-s + ⋯
L(s)  = 1  + (0.852 + 0.521i)2-s − 0.577i·3-s + (0.455 + 0.890i)4-s + 1.44·5-s + (0.301 − 0.492i)6-s + 0.369i·7-s + (−0.0765 + 0.997i)8-s − 0.333·9-s + (1.23 + 0.755i)10-s − 0.987i·11-s + (0.514 − 0.262i)12-s − 0.406·13-s + (−0.193 + 0.315i)14-s − 0.835i·15-s + (−0.585 + 0.810i)16-s − 1.53·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.890 - 0.455i$
Analytic conductor: \(4.88854\)
Root analytic conductor: \(2.21100\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :4),\ 0.890 - 0.455i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.63097 + 0.633426i\)
\(L(\frac12)\) \(\approx\) \(2.63097 + 0.633426i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-13.6 - 8.35i)T \)
3 \( 1 + 46.7iT \)
good5 \( 1 - 904.T + 3.90e5T^{2} \)
7 \( 1 - 888. iT - 5.76e6T^{2} \)
11 \( 1 + 1.44e4iT - 2.14e8T^{2} \)
13 \( 1 + 1.16e4T + 8.15e8T^{2} \)
17 \( 1 + 1.28e5T + 6.97e9T^{2} \)
19 \( 1 + 2.25e5iT - 1.69e10T^{2} \)
23 \( 1 - 4.41e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.15e5T + 5.00e11T^{2} \)
31 \( 1 - 2.01e5iT - 8.52e11T^{2} \)
37 \( 1 - 2.01e6T + 3.51e12T^{2} \)
41 \( 1 + 2.61e6T + 7.98e12T^{2} \)
43 \( 1 + 1.75e6iT - 1.16e13T^{2} \)
47 \( 1 + 4.81e6iT - 2.38e13T^{2} \)
53 \( 1 - 1.23e6T + 6.22e13T^{2} \)
59 \( 1 - 1.14e6iT - 1.46e14T^{2} \)
61 \( 1 + 4.60e6T + 1.91e14T^{2} \)
67 \( 1 - 2.33e7iT - 4.06e14T^{2} \)
71 \( 1 - 3.21e7iT - 6.45e14T^{2} \)
73 \( 1 - 4.89e7T + 8.06e14T^{2} \)
79 \( 1 + 4.23e6iT - 1.51e15T^{2} \)
83 \( 1 - 1.99e7iT - 2.25e15T^{2} \)
89 \( 1 - 8.45e6T + 3.93e15T^{2} \)
97 \( 1 + 8.74e7T + 7.83e15T^{2} \)
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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

−17.95969005248901994080600242231, −17.12521331578421232757569602748, −15.41043311668607877073888510025, −13.73565341430217107141294951610, −13.23235856553115165899059322779, −11.37301366314951607181608921096, −8.922081436105566884733014418443, −6.74656992719755193580773944616, −5.43283531158378359482501058637, −2.43135289177607163127032926336, 2.14170964409159608157572734762, 4.58110670046201228427021405920, 6.29347885965188173133037257070, 9.636839998993331534292908737004, 10.58430648197745025048841818200, 12.57418026455988057495594246976, 13.89890854273163580486540100559, 14.96753764522878578040816840152, 16.73707506637825076448069838137, 18.18971838411060869657096029271

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