Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $0.822 + 0.568i$
Motivic weight 15
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (83.3 − 160. i)2-s + (−3.55e3 + 1.30e3i)3-s + (−1.88e4 − 2.67e4i)4-s + 9.33e4i·5-s + (−8.62e4 + 6.80e5i)6-s + 1.14e6i·7-s + (−5.87e6 + 8.04e5i)8-s + (1.09e7 − 9.29e6i)9-s + (1.50e7 + 7.77e6i)10-s + 1.80e6·11-s + (1.02e8 + 7.05e7i)12-s + 3.17e8·13-s + (1.84e8 + 9.54e7i)14-s + (−1.21e8 − 3.31e8i)15-s + (−3.60e8 + 1.01e9i)16-s − 2.67e9i·17-s + ⋯
L(s)  = 1  + (0.460 − 0.887i)2-s + (−0.938 + 0.344i)3-s + (−0.576 − 0.817i)4-s + 0.534i·5-s + (−0.125 + 0.992i)6-s + 0.525i·7-s + (−0.990 + 0.135i)8-s + (0.761 − 0.647i)9-s + (0.474 + 0.245i)10-s + 0.0279·11-s + (0.822 + 0.568i)12-s + 1.40·13-s + (0.466 + 0.241i)14-s + (−0.184 − 0.501i)15-s + (−0.335 + 0.942i)16-s − 1.58i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(16-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $0.822 + 0.568i$
motivic weight  =  \(15\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :15/2),\ 0.822 + 0.568i)$
$L(8)$  $\approx$  $1.56865 - 0.488944i$
$L(\frac12)$  $\approx$  $1.56865 - 0.488944i$
$L(\frac{17}{2})$   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 \( 1 + (-83.3 + 160. i)T \)
3 \( 1 + (3.55e3 - 1.30e3i)T \)
good5 \( 1 - 9.33e4iT - 3.05e10T^{2} \)
7 \( 1 - 1.14e6iT - 4.74e12T^{2} \)
11 \( 1 - 1.80e6T + 4.17e15T^{2} \)
13 \( 1 - 3.17e8T + 5.11e16T^{2} \)
17 \( 1 + 2.67e9iT - 2.86e18T^{2} \)
19 \( 1 - 2.07e9iT - 1.51e19T^{2} \)
23 \( 1 - 2.43e10T + 2.66e20T^{2} \)
29 \( 1 - 4.77e9iT - 8.62e21T^{2} \)
31 \( 1 - 5.30e10iT - 2.34e22T^{2} \)
37 \( 1 - 4.06e11T + 3.33e23T^{2} \)
41 \( 1 - 1.36e12iT - 1.55e24T^{2} \)
43 \( 1 - 2.85e12iT - 3.17e24T^{2} \)
47 \( 1 + 7.27e11T + 1.20e25T^{2} \)
53 \( 1 + 7.05e12iT - 7.31e25T^{2} \)
59 \( 1 + 1.63e13T + 3.65e26T^{2} \)
61 \( 1 - 3.06e13T + 6.02e26T^{2} \)
67 \( 1 + 7.17e13iT - 2.46e27T^{2} \)
71 \( 1 - 4.82e13T + 5.87e27T^{2} \)
73 \( 1 + 9.56e13T + 8.90e27T^{2} \)
79 \( 1 + 3.27e14iT - 2.91e28T^{2} \)
83 \( 1 - 2.88e14T + 6.11e28T^{2} \)
89 \( 1 + 2.05e14iT - 1.74e29T^{2} \)
97 \( 1 + 5.41e14T + 6.33e29T^{2} \)
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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

−16.14129407084186597165660411202, −14.78324031399392503941153598244, −13.12439618735915539656739891377, −11.65524582151033689503177700365, −10.80896673245218678524678187408, −9.296557686221597940065608331389, −6.36638212607437327563893510691, −4.93412905666406630679758658646, −3.15653429637685260959500703716, −0.997045603762658054585504930147, 0.898348320189514778052369284822, 4.10793344690011006651519333492, 5.62389314824031548796867522614, 6.93261108959974899590168874086, 8.590569804608623272596445337617, 10.91466956548332389298588688700, 12.63131341375103406775666959704, 13.48965082579827465132428967100, 15.38537491374520378398966152357, 16.66063046057353768586739607174

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