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Properties

Label 2-12-3.2-c14-0-4
Degree $2$
Conductor $12$
Sign $-0.783 + 0.620i$
Analytic cond. $14.9194$
Root an. cond. $3.86257$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.71e3 − 1.35e3i)3-s − 9.73e4i·5-s − 5.26e5·7-s + (1.09e6 − 4.65e6i)9-s + 1.93e7i·11-s − 3.32e7·13-s + (−1.32e8 − 1.66e8i)15-s − 5.75e8i·17-s − 1.45e9·19-s + (−9.02e8 + 7.14e8i)21-s − 4.29e9i·23-s − 3.36e9·25-s + (−4.44e9 − 9.46e9i)27-s + 1.71e10i·29-s + 3.40e10·31-s + ⋯
L(s)  = 1  + (0.783 − 0.620i)3-s − 1.24i·5-s − 0.639·7-s + (0.228 − 0.973i)9-s + 0.993i·11-s − 0.530·13-s + (−0.773 − 0.976i)15-s − 1.40i·17-s − 1.62·19-s + (−0.500 + 0.396i)21-s − 1.26i·23-s − 0.551·25-s + (−0.425 − 0.905i)27-s + 0.995i·29-s + 1.23·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.620i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.783 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.783 + 0.620i$
Analytic conductor: \(14.9194\)
Root analytic conductor: \(3.86257\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :7),\ -0.783 + 0.620i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.546755 - 1.57071i\)
\(L(\frac12)\) \(\approx\) \(0.546755 - 1.57071i\)
\(L(8)\) 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 \( 1 + (-1.71e3 + 1.35e3i)T \)
good5 \( 1 + 9.73e4iT - 6.10e9T^{2} \)
7 \( 1 + 5.26e5T + 6.78e11T^{2} \)
11 \( 1 - 1.93e7iT - 3.79e14T^{2} \)
13 \( 1 + 3.32e7T + 3.93e15T^{2} \)
17 \( 1 + 5.75e8iT - 1.68e17T^{2} \)
19 \( 1 + 1.45e9T + 7.99e17T^{2} \)
23 \( 1 + 4.29e9iT - 1.15e19T^{2} \)
29 \( 1 - 1.71e10iT - 2.97e20T^{2} \)
31 \( 1 - 3.40e10T + 7.56e20T^{2} \)
37 \( 1 - 8.31e10T + 9.01e21T^{2} \)
41 \( 1 + 2.11e11iT - 3.79e22T^{2} \)
43 \( 1 - 1.48e11T + 7.38e22T^{2} \)
47 \( 1 - 4.40e11iT - 2.56e23T^{2} \)
53 \( 1 + 1.70e12iT - 1.37e24T^{2} \)
59 \( 1 - 1.79e12iT - 6.19e24T^{2} \)
61 \( 1 - 5.48e12T + 9.87e24T^{2} \)
67 \( 1 - 1.01e13T + 3.67e25T^{2} \)
71 \( 1 - 1.61e12iT - 8.27e25T^{2} \)
73 \( 1 + 8.08e12T + 1.22e26T^{2} \)
79 \( 1 - 1.32e13T + 3.68e26T^{2} \)
83 \( 1 + 8.28e12iT - 7.36e26T^{2} \)
89 \( 1 + 5.62e13iT - 1.95e27T^{2} \)
97 \( 1 + 1.34e12T + 6.52e27T^{2} \)
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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

−16.10718296923772113600009098994, −14.58244799752907580769727372132, −12.98165708748667407510860877144, −12.33986700207610982474666801523, −9.678053229493730828684481089117, −8.511662838536445449533398278643, −6.85508833436717884479189791366, −4.55672099594406924624946629760, −2.37539328154355663317997818718, −0.59052087269575093160410042875, 2.54569530200173737761816667678, 3.80573945038187321695906676205, 6.31956678780733494886625692395, 8.160110574231045732073828342687, 9.887451366340574418945944882436, 10.98645025847358723175777641962, 13.25327148780974804921590455035, 14.58464009222656308802934068409, 15.48495554683527233049002470225, 17.05906905319087775967794921593

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