Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $0.555 - 0.831i$
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 + (−14.6 + 1.03i)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 + (23.0 − 34.5i)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 + (−0.997 + 0.0706i)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 + (0.555 − 0.831i)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

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $0.555 - 0.831i$
motivic weight  =  \(3\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :3/2),\ 0.555 - 0.831i)$
$L(2)$  $\approx$  $0.710465 + 0.380053i$
$L(\frac12)$  $\approx$  $0.710465 + 0.380053i$
$L(\frac{5}{2})$   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

−20.04615591368134888172286197739, −18.64062128153949712611853776322, −16.89791796292965165676381253160, −16.05181287365442182297584093573, −14.75833816652542838209323584236, −13.33606714947912092173490198699, −10.53836900377380684539559835725, −9.171279628502259032815510915885, −7.79450003352070673570454316107, −4.99667685049821096552388374399, 2.75108982179113338574342640129, 7.25318987538709189929739072122, 8.826431911939698895359179639394, 10.63583717773865906397103834563, 12.30827082394096717786522346978, 13.61236202681288569750388567473, 15.30865368685834500042340144278, 17.51155180117663029412240145959, 18.57519266735791446315068748718, 19.31859888442374478632059986411

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