Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $0.888 + 0.459i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 − 2.23i)2-s + (−3.46 + 3.87i)3-s + (−2.00 − 7.74i)4-s + 8.94i·5-s + (2.66 + 14.4i)6-s − 7.74i·7-s + (−20.7 − 8.94i)8-s + (−3.00 − 26.8i)9-s + (20.0 + 15.4i)10-s + 34.6·11-s + (36.9 + 19.0i)12-s − 10·13-s + (−17.3 − 13.4i)14-s + (−34.6 − 30.9i)15-s + (−56 + 30.9i)16-s + 35.7i·17-s + ⋯
L(s)  = 1  + (0.612 − 0.790i)2-s + (−0.666 + 0.745i)3-s + (−0.250 − 0.968i)4-s + 0.799i·5-s + (0.181 + 0.983i)6-s − 0.418i·7-s + (−0.918 − 0.395i)8-s + (−0.111 − 0.993i)9-s + (0.632 + 0.489i)10-s + 0.949·11-s + (0.888 + 0.459i)12-s − 0.213·13-s + (−0.330 − 0.256i)14-s + (−0.596 − 0.533i)15-s + (−0.875 + 0.484i)16-s + 0.510i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $0.888 + 0.459i$
motivic weight  =  \(3\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :3/2),\ 0.888 + 0.459i)$
$L(2)$  $\approx$  $0.972332 - 0.236425i$
$L(\frac12)$  $\approx$  $0.972332 - 0.236425i$
$L(\frac{5}{2})$   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 \( 1 + (-1.73 + 2.23i)T \)
3 \( 1 + (3.46 - 3.87i)T \)
good5 \( 1 - 8.94iT - 125T^{2} \)
7 \( 1 + 7.74iT - 343T^{2} \)
11 \( 1 - 34.6T + 1.33e3T^{2} \)
13 \( 1 + 10T + 2.19e3T^{2} \)
17 \( 1 - 35.7iT - 4.91e3T^{2} \)
19 \( 1 - 69.7iT - 6.85e3T^{2} \)
23 \( 1 + 96.9T + 1.21e4T^{2} \)
29 \( 1 + 152. iT - 2.43e4T^{2} \)
31 \( 1 + 224. iT - 2.97e4T^{2} \)
37 \( 1 + 130T + 5.06e4T^{2} \)
41 \( 1 + 125. iT - 6.89e4T^{2} \)
43 \( 1 - 224. iT - 7.95e4T^{2} \)
47 \( 1 + 193.T + 1.03e5T^{2} \)
53 \( 1 - 545. iT - 1.48e5T^{2} \)
59 \( 1 - 173.T + 2.05e5T^{2} \)
61 \( 1 + 442T + 2.26e5T^{2} \)
67 \( 1 + 735. iT - 3.00e5T^{2} \)
71 \( 1 - 1.03e3T + 3.57e5T^{2} \)
73 \( 1 - 410T + 3.89e5T^{2} \)
79 \( 1 - 85.2iT - 4.93e5T^{2} \)
83 \( 1 + 1.25e3T + 5.71e5T^{2} \)
89 \( 1 - 840. iT - 7.04e5T^{2} \)
97 \( 1 - 770T + 9.12e5T^{2} \)
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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

−19.96227817988236175426714493795, −18.52855858576108479790220107579, −17.01298372103877452068597504567, −15.21922772648573504637430320642, −14.14424290558261059727321999050, −12.13622797696963170452285217551, −10.90848303310514202961708056084, −9.765570252606697516390225853530, −6.19761641252313747646880668290, −3.96721015998628628573876114770, 5.11970921045757825472644203448, 6.84183802607566256049597967972, 8.710033085418608479691832702699, 11.78638494336544608530849049629, 12.74846519852760496323923348560, 14.14962300175953305361265734246, 15.98956461369380405702956087309, 17.02781762442769073291185344769, 18.12779904782448996444243581822, 19.91408288706256799828994702466

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