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Properties

Label 2-12-12.11-c13-0-20
Degree $2$
Conductor $12$
Sign $-0.865 - 0.500i$
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  + (−61.4 − 66.4i)2-s + (544. − 1.13e3i)3-s + (−638. + 8.16e3i)4-s − 3.08e4i·5-s + (−1.09e5 + 3.38e4i)6-s − 3.63e5i·7-s + (5.81e5 − 4.59e5i)8-s + (−1.00e6 − 1.24e6i)9-s + (−2.05e6 + 1.89e6i)10-s + 7.18e6·11-s + (8.95e6 + 5.17e6i)12-s − 2.72e7·13-s + (−2.41e7 + 2.23e7i)14-s + (−3.51e7 − 1.68e7i)15-s + (−6.62e7 − 1.04e7i)16-s + 9.89e7i·17-s + ⋯
L(s)  = 1  + (−0.679 − 0.734i)2-s + (0.431 − 0.902i)3-s + (−0.0779 + 0.996i)4-s − 0.884i·5-s + (−0.955 + 0.296i)6-s − 1.16i·7-s + (0.784 − 0.619i)8-s + (−0.628 − 0.778i)9-s + (−0.649 + 0.600i)10-s + 1.22·11-s + (0.865 + 0.500i)12-s − 1.56·13-s + (−0.856 + 0.792i)14-s + (−0.797 − 0.381i)15-s + (−0.987 − 0.155i)16-s + 0.994i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.865 - 0.500i$
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: \((2,\ 12,\ (\ :13/2),\ -0.865 - 0.500i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.270061 + 1.00731i\)
\(L(\frac12)\) \(\approx\) \(0.270061 + 1.00731i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (61.4 + 66.4i)T \)
3 \( 1 + (-544. + 1.13e3i)T \)
good5 \( 1 + 3.08e4iT - 1.22e9T^{2} \)
7 \( 1 + 3.63e5iT - 9.68e10T^{2} \)
11 \( 1 - 7.18e6T + 3.45e13T^{2} \)
13 \( 1 + 2.72e7T + 3.02e14T^{2} \)
17 \( 1 - 9.89e7iT - 9.90e15T^{2} \)
19 \( 1 - 1.64e8iT - 4.20e16T^{2} \)
23 \( 1 + 1.21e8T + 5.04e17T^{2} \)
29 \( 1 + 3.65e9iT - 1.02e19T^{2} \)
31 \( 1 - 7.81e7iT - 2.44e19T^{2} \)
37 \( 1 + 1.32e10T + 2.43e20T^{2} \)
41 \( 1 + 2.10e10iT - 9.25e20T^{2} \)
43 \( 1 + 2.36e10iT - 1.71e21T^{2} \)
47 \( 1 + 5.28e10T + 5.46e21T^{2} \)
53 \( 1 + 5.07e10iT - 2.60e22T^{2} \)
59 \( 1 - 4.66e11T + 1.04e23T^{2} \)
61 \( 1 - 6.04e11T + 1.61e23T^{2} \)
67 \( 1 + 6.66e11iT - 5.48e23T^{2} \)
71 \( 1 - 1.30e12T + 1.16e24T^{2} \)
73 \( 1 + 2.10e12T + 1.67e24T^{2} \)
79 \( 1 - 1.10e12iT - 4.66e24T^{2} \)
83 \( 1 + 3.62e12T + 8.87e24T^{2} \)
89 \( 1 + 5.87e12iT - 2.19e25T^{2} \)
97 \( 1 - 5.06e11T + 6.73e25T^{2} \)
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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

−16.94238751968647428086663340188, −14.27121291745861605636331134418, −12.86300177011575233602822379229, −11.90723241538989563067929321436, −9.893155142258270453310778465270, −8.438091595745316882425265731560, −7.10855243138077858554488623992, −3.93866438146260751329722774658, −1.75162566912036152296258946050, −0.53027185514772501689040737133, 2.57829615246637249696457715970, 5.08462196961970228176688526327, 6.96482307235058199775407977831, 8.872130120891745327102760071804, 9.846748465317200020761860640311, 11.48603806414302749803594854403, 14.40275903301473334899846054014, 14.91118128815413788180554563124, 16.19254025494089494163184350701, 17.59947475331516677404216174262

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