Properties

Degree 2
Conductor $ 2 \cdot 3 $
Sign $0.336 + 0.941i$
Motivic weight 16
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 181. i·2-s + (6.17e3 − 2.20e3i)3-s − 3.27e4·4-s − 5.48e5i·5-s + (3.99e5 + 1.11e6i)6-s − 8.81e6·7-s − 5.93e6i·8-s + (3.32e7 − 2.72e7i)9-s + 9.92e7·10-s − 2.97e8i·11-s + (−2.02e8 + 7.23e7i)12-s + 6.77e8·13-s − 1.59e9i·14-s + (−1.21e9 − 3.38e9i)15-s + 1.07e9·16-s + 3.75e9i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.941 − 0.336i)3-s − 0.500·4-s − 1.40i·5-s + (0.238 + 0.665i)6-s − 1.52·7-s − 0.353i·8-s + (0.773 − 0.634i)9-s + 0.992·10-s − 1.38i·11-s + (−0.470 + 0.168i)12-s + 0.830·13-s − 1.08i·14-s + (−0.472 − 1.32i)15-s + 0.250·16-s + 0.538i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(17-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6\)    =    \(2 \cdot 3\)
\( \varepsilon \)  =  $0.336 + 0.941i$
motivic weight  =  \(16\)
character  :  $\chi_{6} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6,\ (\ :8),\ 0.336 + 0.941i)$
$L(\frac{17}{2})$  $\approx$  $1.38795 - 0.977760i$
$L(\frac12)$  $\approx$  $1.38795 - 0.977760i$
$L(9)$   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 \( 1 - 181. iT \)
3 \( 1 + (-6.17e3 + 2.20e3i)T \)
good5 \( 1 + 5.48e5iT - 1.52e11T^{2} \)
7 \( 1 + 8.81e6T + 3.32e13T^{2} \)
11 \( 1 + 2.97e8iT - 4.59e16T^{2} \)
13 \( 1 - 6.77e8T + 6.65e17T^{2} \)
17 \( 1 - 3.75e9iT - 4.86e19T^{2} \)
19 \( 1 + 9.70e8T + 2.88e20T^{2} \)
23 \( 1 - 2.68e10iT - 6.13e21T^{2} \)
29 \( 1 + 3.75e11iT - 2.50e23T^{2} \)
31 \( 1 + 4.78e11T + 7.27e23T^{2} \)
37 \( 1 - 9.79e11T + 1.23e25T^{2} \)
41 \( 1 - 1.07e13iT - 6.37e25T^{2} \)
43 \( 1 - 8.94e12T + 1.36e26T^{2} \)
47 \( 1 + 2.80e13iT - 5.66e26T^{2} \)
53 \( 1 - 7.34e13iT - 3.87e27T^{2} \)
59 \( 1 + 1.35e14iT - 2.15e28T^{2} \)
61 \( 1 - 4.44e13T + 3.67e28T^{2} \)
67 \( 1 - 6.70e14T + 1.64e29T^{2} \)
71 \( 1 + 6.75e14iT - 4.16e29T^{2} \)
73 \( 1 - 4.82e14T + 6.50e29T^{2} \)
79 \( 1 - 3.05e14T + 2.30e30T^{2} \)
83 \( 1 - 1.62e15iT - 5.07e30T^{2} \)
89 \( 1 + 3.89e15iT - 1.54e31T^{2} \)
97 \( 1 - 1.41e16T + 6.14e31T^{2} \)
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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

−18.82310474221468497564362417951, −16.64234089813135749250176796230, −15.73428745662231062758418438088, −13.59127463853632432735875182465, −12.81505850937525346743821105561, −9.353985126093516907070868432293, −8.316903820859656618193592963459, −6.13930717651991674205615488518, −3.63165854064562074591928381867, −0.74879565564076440136694501021, 2.49031371516011271071657554530, 3.66337044170614071331562990011, 6.99688512224407447495996652940, 9.450931552343509865561183915468, 10.53725329518493237302244383137, 12.86408880357099527148798080929, 14.35699216036824260266236093852, 15.74917290751665689999918988256, 18.32050377136498189440019602331, 19.32878555191323513607667324831

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