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Properties

Label 2-12-3.2-c12-0-3
Degree $2$
Conductor $12$
Sign $0.818 + 0.574i$
Analytic cond. $10.9679$
Root an. cond. $3.31178$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (596. + 418. i)3-s − 2.39e4i·5-s + 3.21e4·7-s + (1.80e5 + 4.99e5i)9-s − 2.66e6i·11-s + 7.50e6·13-s + (1.00e7 − 1.42e7i)15-s − 1.32e6i·17-s + 3.25e7·19-s + (1.91e7 + 1.34e7i)21-s − 8.48e7i·23-s − 3.27e8·25-s + (−1.01e8 + 3.73e8i)27-s + 8.40e8i·29-s − 1.20e9·31-s + ⋯
L(s)  = 1  + (0.818 + 0.574i)3-s − 1.53i·5-s + 0.273·7-s + (0.340 + 0.940i)9-s − 1.50i·11-s + 1.55·13-s + (0.878 − 1.25i)15-s − 0.0550i·17-s + 0.692·19-s + (0.223 + 0.156i)21-s − 0.573i·23-s − 1.34·25-s + (−0.261 + 0.965i)27-s + 1.41i·29-s − 1.35·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.818 + 0.574i$
Analytic conductor: \(10.9679\)
Root analytic conductor: \(3.31178\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :6),\ 0.818 + 0.574i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.28761 - 0.722597i\)
\(L(\frac12)\) \(\approx\) \(2.28761 - 0.722597i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-596. - 418. i)T \)
good5 \( 1 + 2.39e4iT - 2.44e8T^{2} \)
7 \( 1 - 3.21e4T + 1.38e10T^{2} \)
11 \( 1 + 2.66e6iT - 3.13e12T^{2} \)
13 \( 1 - 7.50e6T + 2.32e13T^{2} \)
17 \( 1 + 1.32e6iT - 5.82e14T^{2} \)
19 \( 1 - 3.25e7T + 2.21e15T^{2} \)
23 \( 1 + 8.48e7iT - 2.19e16T^{2} \)
29 \( 1 - 8.40e8iT - 3.53e17T^{2} \)
31 \( 1 + 1.20e9T + 7.87e17T^{2} \)
37 \( 1 - 1.09e9T + 6.58e18T^{2} \)
41 \( 1 + 5.50e9iT - 2.25e19T^{2} \)
43 \( 1 + 6.27e9T + 3.99e19T^{2} \)
47 \( 1 - 4.76e9iT - 1.16e20T^{2} \)
53 \( 1 - 2.32e10iT - 4.91e20T^{2} \)
59 \( 1 - 1.03e10iT - 1.77e21T^{2} \)
61 \( 1 - 4.64e10T + 2.65e21T^{2} \)
67 \( 1 - 3.54e10T + 8.18e21T^{2} \)
71 \( 1 - 2.45e11iT - 1.64e22T^{2} \)
73 \( 1 - 2.38e11T + 2.29e22T^{2} \)
79 \( 1 + 3.79e10T + 5.90e22T^{2} \)
83 \( 1 + 2.26e11iT - 1.06e23T^{2} \)
89 \( 1 - 8.22e11iT - 2.46e23T^{2} \)
97 \( 1 - 3.60e11T + 6.93e23T^{2} \)
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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

−16.51371031202340341748759737916, −15.97335930603109379057611139052, −14.06816039513934162364115675460, −13.01045219411536386971527689171, −11.02877440858712849701657692778, −9.016742336140603153977658602798, −8.334749112955091691914967090928, −5.34813006668176444665903711633, −3.65436312165417521363218062982, −1.17178079499446013424722990402, 1.89921990853243694337506544887, 3.51871989285637970198923920684, 6.57766638342943031412360853813, 7.78670909119685474590759549832, 9.768533588271037164959740044709, 11.43008669210711750353767530388, 13.27582981788850780895721285447, 14.52952301220362709771189679616, 15.41147007042345105588156094315, 17.94303116205687321002824936557

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