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Properties

Label 2-12-12.11-c15-0-9
Degree $2$
Conductor $12$
Sign $-0.985 + 0.170i$
Analytic cond. $17.1232$
Root an. cond. $4.13802$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (80.5 + 162. i)2-s + (2.76e3 + 2.58e3i)3-s + (−1.97e4 + 2.61e4i)4-s + 3.32e5i·5-s + (−1.96e5 + 6.57e5i)6-s − 2.13e6i·7-s + (−5.82e6 − 1.09e6i)8-s + (9.67e5 + 1.43e7i)9-s + (−5.39e7 + 2.68e7i)10-s + 7.24e7·11-s + (−1.22e8 + 2.11e7i)12-s + 5.97e7·13-s + (3.46e8 − 1.72e8i)14-s + (−8.60e8 + 9.20e8i)15-s + (−2.91e8 − 1.03e9i)16-s − 9.88e8i·17-s + ⋯
L(s)  = 1  + (0.445 + 0.895i)2-s + (0.730 + 0.682i)3-s + (−0.603 + 0.797i)4-s + 1.90i·5-s + (−0.286 + 0.958i)6-s − 0.979i·7-s + (−0.982 − 0.185i)8-s + (0.0674 + 0.997i)9-s + (−1.70 + 0.847i)10-s + 1.12·11-s + (−0.985 + 0.170i)12-s + 0.264·13-s + (0.877 − 0.436i)14-s + (−1.30 + 1.39i)15-s + (−0.271 − 0.962i)16-s − 0.584i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.170i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.985 + 0.170i$
Analytic conductor: \(17.1232\)
Root analytic conductor: \(4.13802\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :15/2),\ -0.985 + 0.170i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.228375 - 2.66086i\)
\(L(\frac12)\) \(\approx\) \(0.228375 - 2.66086i\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-80.5 - 162. i)T \)
3 \( 1 + (-2.76e3 - 2.58e3i)T \)
good5 \( 1 - 3.32e5iT - 3.05e10T^{2} \)
7 \( 1 + 2.13e6iT - 4.74e12T^{2} \)
11 \( 1 - 7.24e7T + 4.17e15T^{2} \)
13 \( 1 - 5.97e7T + 5.11e16T^{2} \)
17 \( 1 + 9.88e8iT - 2.86e18T^{2} \)
19 \( 1 - 6.22e7iT - 1.51e19T^{2} \)
23 \( 1 - 2.88e9T + 2.66e20T^{2} \)
29 \( 1 + 3.09e10iT - 8.62e21T^{2} \)
31 \( 1 - 4.97e10iT - 2.34e22T^{2} \)
37 \( 1 - 2.95e11T + 3.33e23T^{2} \)
41 \( 1 - 1.49e12iT - 1.55e24T^{2} \)
43 \( 1 - 2.69e12iT - 3.17e24T^{2} \)
47 \( 1 - 4.45e12T + 1.20e25T^{2} \)
53 \( 1 + 3.63e12iT - 7.31e25T^{2} \)
59 \( 1 + 2.17e13T + 3.65e26T^{2} \)
61 \( 1 + 3.26e12T + 6.02e26T^{2} \)
67 \( 1 + 7.11e13iT - 2.46e27T^{2} \)
71 \( 1 - 4.18e13T + 5.87e27T^{2} \)
73 \( 1 - 1.11e14T + 8.90e27T^{2} \)
79 \( 1 - 1.98e14iT - 2.91e28T^{2} \)
83 \( 1 + 1.29e14T + 6.11e28T^{2} \)
89 \( 1 + 1.60e14iT - 1.74e29T^{2} \)
97 \( 1 + 1.03e15T + 6.33e29T^{2} \)
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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

−16.83307996303808121763309719337, −15.36948483897217420010560431586, −14.38069369699050546359652972463, −13.77749965715620016558103969482, −11.13410709047506624116070102302, −9.644393327690475565604570854649, −7.65899915038901792152188920961, −6.53607214666335500257099234593, −4.10526212406021671012584967932, −3.03556287492457910144864646842, 0.870567588653024221607501967607, 1.94803837266857507645134763871, 4.00865695449447753694982244641, 5.73462320251582310792721489580, 8.651928171554073564135601514770, 9.223192979358329566485589749080, 11.99698709105116712095391004676, 12.58615659491192388904057473675, 13.78866050627059083089253622980, 15.33585855234767112396250144007

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