Properties

Label 2-12-12.11-c13-0-7
Degree $2$
Conductor $12$
Sign $0.565 - 0.825i$
Analytic cond. $12.8677$
Root an. cond. $3.58715$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−89.8 − 11.2i)2-s + (−433. + 1.18e3i)3-s + (7.93e3 + 2.02e3i)4-s − 2.49e4i·5-s + (5.23e4 − 1.01e5i)6-s − 7.13e4i·7-s + (−6.89e5 − 2.71e5i)8-s + (−1.21e6 − 1.02e6i)9-s + (−2.82e5 + 2.24e6i)10-s + 4.95e6·11-s + (−5.84e6 + 8.53e6i)12-s + 1.35e7·13-s + (−8.05e5 + 6.40e6i)14-s + (2.96e7 + 1.08e7i)15-s + (5.88e7 + 3.22e7i)16-s + 5.28e7i·17-s + ⋯
L(s)  = 1  + (−0.992 − 0.124i)2-s + (−0.343 + 0.939i)3-s + (0.968 + 0.247i)4-s − 0.715i·5-s + (0.457 − 0.889i)6-s − 0.229i·7-s + (−0.930 − 0.366i)8-s + (−0.764 − 0.644i)9-s + (−0.0892 + 0.709i)10-s + 0.842·11-s + (−0.565 + 0.825i)12-s + 0.777·13-s + (−0.0285 + 0.227i)14-s + (0.671 + 0.245i)15-s + (0.877 + 0.479i)16-s + 0.531i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.825i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.565 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.565 - 0.825i$
Analytic conductor: \(12.8677\)
Root analytic conductor: \(3.58715\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :13/2),\ 0.565 - 0.825i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.865010 + 0.455971i\)
\(L(\frac12)\) \(\approx\) \(0.865010 + 0.455971i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (89.8 + 11.2i)T \)
3 \( 1 + (433. - 1.18e3i)T \)
good5 \( 1 + 2.49e4iT - 1.22e9T^{2} \)
7 \( 1 + 7.13e4iT - 9.68e10T^{2} \)
11 \( 1 - 4.95e6T + 3.45e13T^{2} \)
13 \( 1 - 1.35e7T + 3.02e14T^{2} \)
17 \( 1 - 5.28e7iT - 9.90e15T^{2} \)
19 \( 1 - 3.46e8iT - 4.20e16T^{2} \)
23 \( 1 + 4.73e8T + 5.04e17T^{2} \)
29 \( 1 - 6.10e9iT - 1.02e19T^{2} \)
31 \( 1 + 6.26e9iT - 2.44e19T^{2} \)
37 \( 1 - 1.95e10T + 2.43e20T^{2} \)
41 \( 1 - 3.71e10iT - 9.25e20T^{2} \)
43 \( 1 - 2.19e10iT - 1.71e21T^{2} \)
47 \( 1 - 9.42e10T + 5.46e21T^{2} \)
53 \( 1 + 1.89e11iT - 2.60e22T^{2} \)
59 \( 1 + 2.89e11T + 1.04e23T^{2} \)
61 \( 1 - 9.25e10T + 1.61e23T^{2} \)
67 \( 1 + 5.54e10iT - 5.48e23T^{2} \)
71 \( 1 - 8.88e11T + 1.16e24T^{2} \)
73 \( 1 + 3.79e11T + 1.67e24T^{2} \)
79 \( 1 - 8.79e11iT - 4.66e24T^{2} \)
83 \( 1 - 3.05e12T + 8.87e24T^{2} \)
89 \( 1 - 7.30e12iT - 2.19e25T^{2} \)
97 \( 1 + 1.15e13T + 6.73e25T^{2} \)
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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

−16.80717875657887117056456203802, −16.34569964653255674765623499787, −14.74681240003157974674187990856, −12.31058796003295997662936301116, −10.93675041118971435269000495456, −9.608231658792675816567607857751, −8.380162283937625936054858204181, −6.06623778570151182720904210719, −3.82317492318841679684200799151, −1.14883866934439428341106468575, 0.74706002287755199738375673213, 2.49205419433044338682052577509, 6.16824875353982231484162442140, 7.26004314245678205376978008245, 8.914170974902630193286984409470, 10.87194216524513796418598949841, 11.91972477205633133895111567337, 13.90473181721977782914680743798, 15.54912977980109000661139911854, 17.12945078862855020031863763019

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