Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $0.226 + 0.974i$
Motivic weight 19
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (183. − 700. i)2-s + (−2.29e4 − 2.51e4i)3-s + (−4.57e5 − 2.56e5i)4-s + 6.84e6i·5-s + (−2.18e7 + 1.14e7i)6-s − 8.10e7i·7-s + (−2.63e8 + 2.73e8i)8-s + (−1.05e8 + 1.15e9i)9-s + (4.79e9 + 1.25e9i)10-s + 1.02e10·11-s + (4.04e9 + 1.74e10i)12-s − 9.13e9·13-s + (−5.67e10 − 1.48e10i)14-s + (1.72e11 − 1.57e11i)15-s + (1.43e11 + 2.34e11i)16-s − 6.29e11i·17-s + ⋯
L(s)  = 1  + (0.253 − 0.967i)2-s + (−0.674 − 0.738i)3-s + (−0.871 − 0.489i)4-s + 1.56i·5-s + (−0.885 + 0.465i)6-s − 0.759i·7-s + (−0.694 + 0.719i)8-s + (−0.0911 + 0.995i)9-s + (1.51 + 0.396i)10-s + 1.31·11-s + (0.226 + 0.974i)12-s − 0.238·13-s + (−0.734 − 0.192i)14-s + (1.15 − 1.05i)15-s + (0.520 + 0.853i)16-s − 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $0.226 + 0.974i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ 0.226 + 0.974i)$
$L(10)$  $\approx$  $1.525178309$
$L(\frac12)$  $\approx$  $1.525178309$
$L(\frac{21}{2})$   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 \( 1 + (-183. + 700. i)T \)
3 \( 1 + (2.29e4 + 2.51e4i)T \)
good5 \( 1 - 6.84e6iT - 1.90e13T^{2} \)
7 \( 1 + 8.10e7iT - 1.13e16T^{2} \)
11 \( 1 - 1.02e10T + 6.11e19T^{2} \)
13 \( 1 + 9.13e9T + 1.46e21T^{2} \)
17 \( 1 + 6.29e11iT - 2.39e23T^{2} \)
19 \( 1 - 2.01e12iT - 1.97e24T^{2} \)
23 \( 1 + 2.86e12T + 7.46e25T^{2} \)
29 \( 1 - 8.26e12iT - 6.10e27T^{2} \)
31 \( 1 + 1.75e14iT - 2.16e28T^{2} \)
37 \( 1 - 1.30e15T + 6.24e29T^{2} \)
41 \( 1 + 5.00e14iT - 4.39e30T^{2} \)
43 \( 1 + 2.29e15iT - 1.08e31T^{2} \)
47 \( 1 - 8.03e15T + 5.88e31T^{2} \)
53 \( 1 - 2.50e16iT - 5.77e32T^{2} \)
59 \( 1 - 6.40e16T + 4.42e33T^{2} \)
61 \( 1 - 4.18e16T + 8.34e33T^{2} \)
67 \( 1 - 8.57e16iT - 4.95e34T^{2} \)
71 \( 1 - 5.20e17T + 1.49e35T^{2} \)
73 \( 1 + 3.77e17T + 2.53e35T^{2} \)
79 \( 1 - 4.09e17iT - 1.13e36T^{2} \)
83 \( 1 - 1.71e18T + 2.90e36T^{2} \)
89 \( 1 - 6.97e17iT - 1.09e37T^{2} \)
97 \( 1 - 8.71e18T + 5.60e37T^{2} \)
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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

−14.51050435449125670253846329080, −13.74852287562148514944476129482, −11.99579602492522416561085644110, −11.10542110019685259283070342860, −9.938511713992336649787968201828, −7.37307945030620058488350107936, −6.06357939260807146802833638911, −3.93581444975384391282443214241, −2.34428566887354664297221337654, −0.812669973658071531447621293473, 0.807984879168604672047683646410, 4.08090277798381087547160672431, 5.09171613569174835029096337733, 6.28947352911024065606545901350, 8.634580325577911965204685554696, 9.389809501635411577094795604195, 11.87571812135882125220172146416, 12.86814538444060508368398299946, 14.79485257363968811023415220337, 15.92208092009948568405003537579

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