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Properties

Label 2-12-12.11-c19-0-5
Degree $2$
Conductor $12$
Sign $0.881 + 0.472i$
Analytic cond. $27.4580$
Root an. cond. $5.24003$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (74.5 − 720. i)2-s + (−3.27e4 − 9.61e3i)3-s + (−5.13e5 − 1.07e5i)4-s − 7.07e6i·5-s + (−9.36e6 + 2.28e7i)6-s + 1.56e8i·7-s + (−1.15e8 + 3.61e8i)8-s + (9.77e8 + 6.29e8i)9-s + (−5.09e9 − 5.26e8i)10-s − 9.54e9·11-s + (1.57e10 + 8.44e9i)12-s − 2.20e10·13-s + (1.12e11 + 1.16e10i)14-s + (−6.80e10 + 2.31e11i)15-s + (2.51e11 + 1.10e11i)16-s − 1.32e11i·17-s + ⋯
L(s)  = 1  + (0.102 − 0.994i)2-s + (−0.959 − 0.282i)3-s + (−0.978 − 0.204i)4-s − 1.61i·5-s + (−0.379 + 0.925i)6-s + 1.46i·7-s + (−0.304 + 0.952i)8-s + (0.840 + 0.541i)9-s + (−1.61 − 0.166i)10-s − 1.22·11-s + (0.881 + 0.472i)12-s − 0.576·13-s + (1.45 + 0.150i)14-s + (−0.456 + 1.55i)15-s + (0.916 + 0.400i)16-s − 0.271i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.881 + 0.472i$
Analytic conductor: \(27.4580\)
Root analytic conductor: \(5.24003\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :19/2),\ 0.881 + 0.472i)\)

Particular Values

\(L(10)\) \(\approx\) \(0.7039667375\)
\(L(\frac12)\) \(\approx\) \(0.7039667375\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-74.5 + 720. i)T \)
3 \( 1 + (3.27e4 + 9.61e3i)T \)
good5 \( 1 + 7.07e6iT - 1.90e13T^{2} \)
7 \( 1 - 1.56e8iT - 1.13e16T^{2} \)
11 \( 1 + 9.54e9T + 6.11e19T^{2} \)
13 \( 1 + 2.20e10T + 1.46e21T^{2} \)
17 \( 1 + 1.32e11iT - 2.39e23T^{2} \)
19 \( 1 - 1.26e11iT - 1.97e24T^{2} \)
23 \( 1 - 1.44e13T + 7.46e25T^{2} \)
29 \( 1 + 5.22e13iT - 6.10e27T^{2} \)
31 \( 1 + 8.91e12iT - 2.16e28T^{2} \)
37 \( 1 + 5.03e14T + 6.24e29T^{2} \)
41 \( 1 - 1.92e15iT - 4.39e30T^{2} \)
43 \( 1 - 1.96e14iT - 1.08e31T^{2} \)
47 \( 1 + 6.24e15T + 5.88e31T^{2} \)
53 \( 1 - 9.61e15iT - 5.77e32T^{2} \)
59 \( 1 - 2.49e16T + 4.42e33T^{2} \)
61 \( 1 - 1.42e17T + 8.34e33T^{2} \)
67 \( 1 - 1.59e17iT - 4.95e34T^{2} \)
71 \( 1 - 6.13e17T + 1.49e35T^{2} \)
73 \( 1 - 4.40e17T + 2.53e35T^{2} \)
79 \( 1 - 1.92e18iT - 1.13e36T^{2} \)
83 \( 1 + 1.68e18T + 2.90e36T^{2} \)
89 \( 1 - 2.75e18iT - 1.09e37T^{2} \)
97 \( 1 - 7.95e18T + 5.60e37T^{2} \)
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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

−15.53982436909595548850939701014, −13.05795983659077265975390573092, −12.51411015952824052505842920668, −11.44733010210079369596691573447, −9.655186032118116618953820218591, −8.348604197500489441814936467780, −5.41123251644394049959403066159, −4.91675892039652334570918702478, −2.32047702388217761657570989060, −0.874912813533297914789816557005, 0.34995636726397858733216380058, 3.47758288283235162339294053670, 5.06061907456107407475570819072, 6.74922015535052836207091939499, 7.38557591660419490826072863125, 10.11915088838096209063192531526, 10.87143315826983058001958114054, 13.08272122286168422711078221178, 14.46616747452579190698223178990, 15.58844690793857932707858119378

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