Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $0.990 - 0.140i$
Motivic weight 19
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (706. − 159. i)2-s + (3.25e4 + 1.01e4i)3-s + (4.73e5 − 2.24e5i)4-s + 5.54e6i·5-s + (2.46e7 + 1.98e6i)6-s − 1.49e8i·7-s + (2.98e8 − 2.34e8i)8-s + (9.56e8 + 6.59e8i)9-s + (8.81e8 + 3.91e9i)10-s + 2.49e6·11-s + (1.76e10 − 2.50e9i)12-s + 4.88e10·13-s + (−2.38e10 − 1.05e11i)14-s + (−5.62e10 + 1.80e11i)15-s + (1.73e11 − 2.12e11i)16-s + 4.51e11i·17-s + ⋯
L(s)  = 1  + (0.975 − 0.219i)2-s + (0.954 + 0.297i)3-s + (0.903 − 0.428i)4-s + 1.26i·5-s + (0.996 + 0.0804i)6-s − 1.40i·7-s + (0.787 − 0.616i)8-s + (0.823 + 0.567i)9-s + (0.278 + 1.23i)10-s + 0.000319·11-s + (0.990 − 0.140i)12-s + 1.27·13-s + (−0.308 − 1.37i)14-s + (−0.377 + 1.21i)15-s + (0.632 − 0.774i)16-s + 0.923i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(20-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $0.990 - 0.140i$
motivic weight  =  \(19\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :19/2),\ 0.990 - 0.140i)$
$L(10)$  $\approx$  $5.503003596$
$L(\frac12)$  $\approx$  $5.503003596$
$L(\frac{21}{2})$   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 \( 1 + (-706. + 159. i)T \)
3 \( 1 + (-3.25e4 - 1.01e4i)T \)
good5 \( 1 - 5.54e6iT - 1.90e13T^{2} \)
7 \( 1 + 1.49e8iT - 1.13e16T^{2} \)
11 \( 1 - 2.49e6T + 6.11e19T^{2} \)
13 \( 1 - 4.88e10T + 1.46e21T^{2} \)
17 \( 1 - 4.51e11iT - 2.39e23T^{2} \)
19 \( 1 - 6.03e11iT - 1.97e24T^{2} \)
23 \( 1 + 7.40e12T + 7.46e25T^{2} \)
29 \( 1 - 1.10e14iT - 6.10e27T^{2} \)
31 \( 1 + 2.18e14iT - 2.16e28T^{2} \)
37 \( 1 - 4.99e14T + 6.24e29T^{2} \)
41 \( 1 - 1.26e13iT - 4.39e30T^{2} \)
43 \( 1 + 3.26e15iT - 1.08e31T^{2} \)
47 \( 1 + 1.12e16T + 5.88e31T^{2} \)
53 \( 1 + 3.23e16iT - 5.77e32T^{2} \)
59 \( 1 + 6.67e16T + 4.42e33T^{2} \)
61 \( 1 + 1.62e17T + 8.34e33T^{2} \)
67 \( 1 + 4.88e15iT - 4.95e34T^{2} \)
71 \( 1 - 2.98e17T + 1.49e35T^{2} \)
73 \( 1 + 9.07e17T + 2.53e35T^{2} \)
79 \( 1 + 3.92e17iT - 1.13e36T^{2} \)
83 \( 1 - 8.04e17T + 2.90e36T^{2} \)
89 \( 1 - 1.37e18iT - 1.09e37T^{2} \)
97 \( 1 + 1.68e18T + 5.60e37T^{2} \)
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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

−14.98029964034958058460064514538, −14.11199664131557877513379822791, −13.21715335566118332973835124330, −10.94856382353364257195850896557, −10.23138773735256273139299139752, −7.72073808504216075193834776636, −6.44455784716791539814448459300, −4.03679004378228702073912595242, −3.31172326824091601136866110502, −1.65119143720292475158381405866, 1.49875799043321831878255979999, 2.90484291594817233886761784775, 4.58857418840610483116148358365, 6.08065349148935565402041389789, 8.096543189769249071265556622478, 9.101924248145209085006346684386, 11.83337515000765848046008565368, 12.83298908913680154702984024068, 13.86429705118786140329488695579, 15.41298070422864152066354389572

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