Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $0.881 - 0.472i$
Motivic weight 19
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

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $0.881 - 0.472i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ 0.881 - 0.472i)$
$L(10)$  $\approx$  $0.7039667375$
$L(\frac12)$  $\approx$  $0.7039667375$
$L(\frac{21}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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