Properties

Degree 2
Conductor $ 2^{2} \cdot 3 $
Sign $-0.103 - 0.994i$
Motivic weight 15
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−80.5 − 162. i)2-s + (−2.76e3 + 2.58e3i)3-s + (−1.97e4 + 2.61e4i)4-s − 3.32e5i·5-s + (6.42e5 + 2.40e5i)6-s − 2.13e6i·7-s + (5.82e6 + 1.09e6i)8-s + (9.67e5 − 1.43e7i)9-s + (−5.39e7 + 2.68e7i)10-s − 7.24e7·11-s + (−1.28e7 − 1.23e8i)12-s + 5.97e7·13-s + (−3.46e8 + 1.72e8i)14-s + (8.60e8 + 9.20e8i)15-s + (−2.91e8 − 1.03e9i)16-s + 9.88e8i·17-s + ⋯
L(s)  = 1  + (−0.445 − 0.895i)2-s + (−0.730 + 0.682i)3-s + (−0.603 + 0.797i)4-s − 1.90i·5-s + (0.936 + 0.350i)6-s − 0.979i·7-s + (0.982 + 0.185i)8-s + (0.0674 − 0.997i)9-s + (−1.70 + 0.847i)10-s − 1.12·11-s + (−0.103 − 0.994i)12-s + 0.264·13-s + (−0.877 + 0.436i)14-s + (1.30 + 1.39i)15-s + (−0.271 − 0.962i)16-s + 0.584i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(16-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12\)    =    \(2^{2} \cdot 3\)
\( \varepsilon \)  =  $-0.103 - 0.994i$
motivic weight  =  \(15\)
character  :  $\chi_{12} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 12,\ (\ :15/2),\ -0.103 - 0.994i)$
$L(8)$  $\approx$  $0.116844 + 0.129641i$
$L(\frac12)$  $\approx$  $0.116844 + 0.129641i$
$L(\frac{17}{2})$   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 \( 1 + (80.5 + 162. i)T \)
3 \( 1 + (2.76e3 - 2.58e3i)T \)
good5 \( 1 + 3.32e5iT - 3.05e10T^{2} \)
7 \( 1 + 2.13e6iT - 4.74e12T^{2} \)
11 \( 1 + 7.24e7T + 4.17e15T^{2} \)
13 \( 1 - 5.97e7T + 5.11e16T^{2} \)
17 \( 1 - 9.88e8iT - 2.86e18T^{2} \)
19 \( 1 - 6.22e7iT - 1.51e19T^{2} \)
23 \( 1 + 2.88e9T + 2.66e20T^{2} \)
29 \( 1 - 3.09e10iT - 8.62e21T^{2} \)
31 \( 1 - 4.97e10iT - 2.34e22T^{2} \)
37 \( 1 - 2.95e11T + 3.33e23T^{2} \)
41 \( 1 + 1.49e12iT - 1.55e24T^{2} \)
43 \( 1 - 2.69e12iT - 3.17e24T^{2} \)
47 \( 1 + 4.45e12T + 1.20e25T^{2} \)
53 \( 1 - 3.63e12iT - 7.31e25T^{2} \)
59 \( 1 - 2.17e13T + 3.65e26T^{2} \)
61 \( 1 + 3.26e12T + 6.02e26T^{2} \)
67 \( 1 + 7.11e13iT - 2.46e27T^{2} \)
71 \( 1 + 4.18e13T + 5.87e27T^{2} \)
73 \( 1 - 1.11e14T + 8.90e27T^{2} \)
79 \( 1 - 1.98e14iT - 2.91e28T^{2} \)
83 \( 1 - 1.29e14T + 6.11e28T^{2} \)
89 \( 1 - 1.60e14iT - 1.74e29T^{2} \)
97 \( 1 + 1.03e15T + 6.33e29T^{2} \)
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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

−16.07543790748843815486032170998, −13.30098945501906141566514617652, −12.35061948618189502157600538080, −10.82049510521838052643460800975, −9.593098734941240476281945848599, −8.180795608003733621496060849511, −5.11660331683327681048126541901, −4.02049767147417055171508632030, −1.17632104744704135824400057571, −0.097357163171033544571664301198, 2.39274647600330702122391840441, 5.57426722707666338030193145137, 6.68201361091114708187084540180, 7.86329536805105307175183836945, 10.17587411650993902472179315093, 11.38270049393665406587036684231, 13.47780517688040816488361716813, 14.87528638729989987508469646403, 15.96253536162385038583736240884, 17.83517790953025507321946990844

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