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Properties

Label 2-12-12.11-c11-0-7
Degree $2$
Conductor $12$
Sign $0.626 - 0.779i$
Analytic cond. $9.22011$
Root an. cond. $3.03646$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (24.2 − 38.2i)2-s + (183. + 378. i)3-s + (−876. − 1.85e3i)4-s + 9.92e3i·5-s + (1.89e4 + 2.13e3i)6-s + 3.59e4i·7-s + (−9.19e4 − 1.12e4i)8-s + (−1.09e5 + 1.39e5i)9-s + (3.79e5 + 2.40e5i)10-s + 1.23e5·11-s + (5.39e5 − 6.71e5i)12-s + 2.19e6·13-s + (1.37e6 + 8.69e5i)14-s + (−3.75e6 + 1.82e6i)15-s + (−2.65e6 + 3.24e6i)16-s + 3.57e6i·17-s + ⋯
L(s)  = 1  + (0.534 − 0.844i)2-s + (0.436 + 0.899i)3-s + (−0.427 − 0.903i)4-s + 1.41i·5-s + (0.993 + 0.112i)6-s + 0.808i·7-s + (−0.992 − 0.121i)8-s + (−0.618 + 0.785i)9-s + (1.19 + 0.759i)10-s + 0.230·11-s + (0.626 − 0.779i)12-s + 1.63·13-s + (0.682 + 0.432i)14-s + (−1.27 + 0.620i)15-s + (−0.633 + 0.773i)16-s + 0.610i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.626 - 0.779i$
Analytic conductor: \(9.22011\)
Root analytic conductor: \(3.03646\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :11/2),\ 0.626 - 0.779i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.06896 + 0.991803i\)
\(L(\frac12)\) \(\approx\) \(2.06896 + 0.991803i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-24.2 + 38.2i)T \)
3 \( 1 + (-183. - 378. i)T \)
good5 \( 1 - 9.92e3iT - 4.88e7T^{2} \)
7 \( 1 - 3.59e4iT - 1.97e9T^{2} \)
11 \( 1 - 1.23e5T + 2.85e11T^{2} \)
13 \( 1 - 2.19e6T + 1.79e12T^{2} \)
17 \( 1 - 3.57e6iT - 3.42e13T^{2} \)
19 \( 1 + 1.28e7iT - 1.16e14T^{2} \)
23 \( 1 + 2.29e7T + 9.52e14T^{2} \)
29 \( 1 - 8.68e7iT - 1.22e16T^{2} \)
31 \( 1 + 1.57e8iT - 2.54e16T^{2} \)
37 \( 1 - 4.22e8T + 1.77e17T^{2} \)
41 \( 1 - 1.60e8iT - 5.50e17T^{2} \)
43 \( 1 + 7.07e8iT - 9.29e17T^{2} \)
47 \( 1 - 3.00e9T + 2.47e18T^{2} \)
53 \( 1 + 1.95e9iT - 9.26e18T^{2} \)
59 \( 1 - 3.47e9T + 3.01e19T^{2} \)
61 \( 1 + 3.58e9T + 4.35e19T^{2} \)
67 \( 1 - 7.20e9iT - 1.22e20T^{2} \)
71 \( 1 + 1.78e10T + 2.31e20T^{2} \)
73 \( 1 - 2.02e9T + 3.13e20T^{2} \)
79 \( 1 + 2.09e10iT - 7.47e20T^{2} \)
83 \( 1 - 3.45e10T + 1.28e21T^{2} \)
89 \( 1 - 6.54e10iT - 2.77e21T^{2} \)
97 \( 1 - 9.06e10T + 7.15e21T^{2} \)
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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

−18.15485829702481654052989572998, −15.60775492320250885901541329013, −14.77510050760261613741311730365, −13.56602107133810277685258128771, −11.39877965295826235426247511333, −10.50705670254518819105389018144, −8.937263548827726259196692118972, −5.98613378391332098314731012597, −3.80243991093397250379868519794, −2.50550691043656735611442710764, 0.986776423586286621383654152971, 3.97129389462501544165290827859, 6.01728403485909657957124812180, 7.80458071569369346747488945892, 8.899696296765202987661786316455, 12.07687945735985279058405737738, 13.24735729886586404035072244820, 14.04726281480966877187497757131, 15.99050248108515010855696011437, 17.01141449169991138029563926024

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