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Properties

Label 2-12-12.11-c7-0-11
Degree $2$
Conductor $12$
Sign $-0.712 + 0.701i$
Analytic cond. $3.74862$
Root an. cond. $1.93613$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.29 − 11.0i)2-s + (17.5 − 43.3i)3-s + (−117. − 50.9i)4-s + 151. i·5-s + (−440. − 293. i)6-s − 893. i·7-s + (−834. + 1.18e3i)8-s + (−1.57e3 − 1.51e3i)9-s + (1.68e3 + 349. i)10-s + 7.69e3·11-s + (−4.26e3 + 4.19e3i)12-s + 5.58e3·13-s + (−9.89e3 − 2.05e3i)14-s + (6.58e3 + 2.66e3i)15-s + (1.11e4 + 1.19e4i)16-s + 3.13e3i·17-s + ⋯
L(s)  = 1  + (0.203 − 0.979i)2-s + (0.374 − 0.927i)3-s + (−0.917 − 0.397i)4-s + 0.543i·5-s + (−0.831 − 0.555i)6-s − 0.984i·7-s + (−0.576 + 0.817i)8-s + (−0.719 − 0.694i)9-s + (0.532 + 0.110i)10-s + 1.74·11-s + (−0.712 + 0.701i)12-s + 0.704·13-s + (−0.963 − 0.200i)14-s + (0.503 + 0.203i)15-s + (0.683 + 0.730i)16-s + 0.155i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.712 + 0.701i$
Analytic conductor: \(3.74862\)
Root analytic conductor: \(1.93613\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :7/2),\ -0.712 + 0.701i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.612819 - 1.49611i\)
\(L(\frac12)\) \(\approx\) \(0.612819 - 1.49611i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.29 + 11.0i)T \)
3 \( 1 + (-17.5 + 43.3i)T \)
good5 \( 1 - 151. iT - 7.81e4T^{2} \)
7 \( 1 + 893. iT - 8.23e5T^{2} \)
11 \( 1 - 7.69e3T + 1.94e7T^{2} \)
13 \( 1 - 5.58e3T + 6.27e7T^{2} \)
17 \( 1 - 3.13e3iT - 4.10e8T^{2} \)
19 \( 1 + 1.71e4iT - 8.93e8T^{2} \)
23 \( 1 + 4.78e4T + 3.40e9T^{2} \)
29 \( 1 - 2.04e5iT - 1.72e10T^{2} \)
31 \( 1 - 1.28e5iT - 2.75e10T^{2} \)
37 \( 1 + 3.24e5T + 9.49e10T^{2} \)
41 \( 1 - 3.22e5iT - 1.94e11T^{2} \)
43 \( 1 + 4.18e5iT - 2.71e11T^{2} \)
47 \( 1 - 6.54e5T + 5.06e11T^{2} \)
53 \( 1 + 5.06e4iT - 1.17e12T^{2} \)
59 \( 1 + 1.93e6T + 2.48e12T^{2} \)
61 \( 1 - 1.37e6T + 3.14e12T^{2} \)
67 \( 1 + 2.50e6iT - 6.06e12T^{2} \)
71 \( 1 - 1.68e5T + 9.09e12T^{2} \)
73 \( 1 + 3.41e5T + 1.10e13T^{2} \)
79 \( 1 - 3.19e6iT - 1.92e13T^{2} \)
83 \( 1 - 2.25e6T + 2.71e13T^{2} \)
89 \( 1 + 4.66e6iT - 4.42e13T^{2} \)
97 \( 1 + 5.57e6T + 8.07e13T^{2} \)
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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

−18.39763107916575485168543721530, −17.22830339059997360735772629649, −14.47297155273885909146091144970, −13.73866719161800363407459301648, −12.21151155607156322797931212696, −10.82862889179139905717838594551, −8.903221668635823666321722487168, −6.73010153761778223638224265107, −3.58888682966595410436067759997, −1.27298276528861622387522654202, 4.06809259066615237080316720318, 5.89611415433032889647094406103, 8.489980765881844648413441421905, 9.424645658624068458107981190406, 12.00297845803585079873438181311, 13.91203244438550973289176237462, 15.06979551006684874696028846742, 16.16818120488784284618193673603, 17.23033249824592485882312234456, 19.00087813329195537337023760020

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