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Properties

Label 2-12-12.11-c13-0-0
Degree $2$
Conductor $12$
Sign $-0.933 + 0.359i$
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 + (4.22e4 − 1.06e5i)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 + (9.65e6 − 3.72e6i)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 + (0.369 − 0.929i)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 + (0.933 − 0.359i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.933 + 0.359i$
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: \((2,\ 12,\ (\ :13/2),\ -0.933 + 0.359i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.0453280 - 0.243624i\)
\(L(\frac12)\) \(\approx\) \(0.0453280 - 0.243624i\)
\(L(\frac{15}{2})\) 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

−17.52312447076684289578386656020, −16.39609103872536400176735323867, −14.58371690642029421245657088822, −13.59788828595324490465486895445, −12.28018947409489693918967464713, −10.62264210411443190656204639589, −7.60827803241636324651635801256, −6.97277087634179604596537952114, −5.11283399637488599238799142624, −2.78501044395381272317221259537, 0.082298909986849175996384490413, 2.56636784096329431960666827457, 4.69872952662406378595058164395, 5.58492750332208017364652027629, 8.976970191333043161361308360540, 10.28644525090287707787025562092, 11.88354667825637635015866939636, 12.84068347859080703435887890098, 14.90575524615779586919761551707, 15.76682697527371919764003965642

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