Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $0.866 + 0.499i$
Motivic weight 15
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−35.8 − 177. i)2-s + (−2.29e3 − 3.01e3i)3-s + (−3.01e4 + 1.27e4i)4-s + 3.34e4i·5-s + (−4.53e5 + 5.14e5i)6-s + 6.94e5i·7-s + (3.33e6 + 4.90e6i)8-s + (−3.84e6 + 1.38e7i)9-s + (5.93e6 − 1.19e6i)10-s + 5.49e7·11-s + (1.07e8 + 6.19e7i)12-s − 2.93e8·13-s + (1.23e8 − 2.48e7i)14-s + (1.00e8 − 7.67e7i)15-s + (7.50e8 − 7.68e8i)16-s + 1.09e9i·17-s + ⋯
L(s)  = 1  + (−0.197 − 0.980i)2-s + (−0.604 − 0.796i)3-s + (−0.921 + 0.388i)4-s + 0.191i·5-s + (−0.660 + 0.750i)6-s + 0.318i·7-s + (0.562 + 0.826i)8-s + (−0.268 + 0.963i)9-s + (0.187 − 0.0379i)10-s + 0.850·11-s + (0.866 + 0.499i)12-s − 1.29·13-s + (0.312 − 0.0630i)14-s + (0.152 − 0.115i)15-s + (0.698 − 0.715i)16-s + 0.644i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(16-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $0.866 + 0.499i$
motivic weight  =  \(15\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :15/2),\ 0.866 + 0.499i)$
$L(8)$  $\approx$  $0.952699 - 0.254695i$
$L(\frac12)$  $\approx$  $0.952699 - 0.254695i$
$L(\frac{17}{2})$   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 \( 1 + (35.8 + 177. i)T \)
3 \( 1 + (2.29e3 + 3.01e3i)T \)
good5 \( 1 - 3.34e4iT - 3.05e10T^{2} \)
7 \( 1 - 6.94e5iT - 4.74e12T^{2} \)
11 \( 1 - 5.49e7T + 4.17e15T^{2} \)
13 \( 1 + 2.93e8T + 5.11e16T^{2} \)
17 \( 1 - 1.09e9iT - 2.86e18T^{2} \)
19 \( 1 + 5.34e9iT - 1.51e19T^{2} \)
23 \( 1 - 1.46e10T + 2.66e20T^{2} \)
29 \( 1 - 7.94e10iT - 8.62e21T^{2} \)
31 \( 1 - 1.27e11iT - 2.34e22T^{2} \)
37 \( 1 + 4.22e11T + 3.33e23T^{2} \)
41 \( 1 - 7.06e11iT - 1.55e24T^{2} \)
43 \( 1 - 2.89e12iT - 3.17e24T^{2} \)
47 \( 1 - 6.40e12T + 1.20e25T^{2} \)
53 \( 1 + 7.99e11iT - 7.31e25T^{2} \)
59 \( 1 - 1.39e13T + 3.65e26T^{2} \)
61 \( 1 - 9.62e12T + 6.02e26T^{2} \)
67 \( 1 - 2.75e13iT - 2.46e27T^{2} \)
71 \( 1 + 1.40e14T + 5.87e27T^{2} \)
73 \( 1 - 1.33e14T + 8.90e27T^{2} \)
79 \( 1 + 1.22e14iT - 2.91e28T^{2} \)
83 \( 1 + 8.03e13T + 6.11e28T^{2} \)
89 \( 1 - 4.85e14iT - 1.74e29T^{2} \)
97 \( 1 - 1.97e14T + 6.33e29T^{2} \)
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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

−16.92271687691589927069574737754, −14.50270030299504282970618902565, −12.94229361472335152019660148295, −11.94783229685313443664652550286, −10.72493842537511447011185554911, −8.931458699386034791122382925174, −7.04865239227501063368745606679, −4.94385670092409187687621404032, −2.63738726901321699291341115987, −1.06346101308968340095286253043, 0.56659344872335834166221963959, 4.11901475533367688619729708218, 5.46868536730378483860854795453, 7.08979459981563484574836000785, 9.074413703290487653187051825908, 10.25066242701062473586635783964, 12.18322549865190004242865193423, 14.20072045084112249750940156529, 15.29201277691557920234786866506, 16.75493480924983689493595446561

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