Properties

Degree 2
Conductor $ 2 \cdot 3 $
Sign $0.257 - 0.966i$
Motivic weight 18
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 362. i·2-s + (−1.90e4 − 5.06e3i)3-s − 1.31e5·4-s − 2.98e6i·5-s + (−1.83e6 + 6.88e6i)6-s − 4.92e7·7-s + 4.74e7i·8-s + (3.36e8 + 1.92e8i)9-s − 1.08e9·10-s + 5.85e8i·11-s + (2.49e9 + 6.64e8i)12-s + 1.21e10·13-s + 1.78e10i·14-s + (−1.51e10 + 5.68e10i)15-s + 1.71e10·16-s + 1.49e11i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.966 − 0.257i)3-s − 0.500·4-s − 1.53i·5-s + (−0.182 + 0.683i)6-s − 1.22·7-s + 0.353i·8-s + (0.867 + 0.497i)9-s − 1.08·10-s + 0.248i·11-s + (0.483 + 0.128i)12-s + 1.14·13-s + 0.863i·14-s + (−0.394 + 1.47i)15-s + 0.250·16-s + 1.26i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(19-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6\)    =    \(2 \cdot 3\)
\( \varepsilon \)  =  $0.257 - 0.966i$
motivic weight  =  \(18\)
character  :  $\chi_{6} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6,\ (\ :9),\ 0.257 - 0.966i)$
$L(\frac{19}{2})$  $\approx$  $0.00457753 + 0.00351748i$
$L(\frac12)$  $\approx$  $0.00457753 + 0.00351748i$
$L(10)$   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 \( 1 + 362. iT \)
3 \( 1 + (1.90e4 + 5.06e3i)T \)
good5 \( 1 + 2.98e6iT - 3.81e12T^{2} \)
7 \( 1 + 4.92e7T + 1.62e15T^{2} \)
11 \( 1 - 5.85e8iT - 5.55e18T^{2} \)
13 \( 1 - 1.21e10T + 1.12e20T^{2} \)
17 \( 1 - 1.49e11iT - 1.40e22T^{2} \)
19 \( 1 + 3.13e11T + 1.04e23T^{2} \)
23 \( 1 + 2.41e12iT - 3.24e24T^{2} \)
29 \( 1 - 1.39e13iT - 2.10e26T^{2} \)
31 \( 1 + 2.16e13T + 6.99e26T^{2} \)
37 \( 1 - 2.89e13T + 1.68e28T^{2} \)
41 \( 1 + 2.60e14iT - 1.07e29T^{2} \)
43 \( 1 + 9.12e13T + 2.52e29T^{2} \)
47 \( 1 - 1.34e15iT - 1.25e30T^{2} \)
53 \( 1 - 4.92e15iT - 1.08e31T^{2} \)
59 \( 1 - 1.63e15iT - 7.50e31T^{2} \)
61 \( 1 + 1.27e16T + 1.36e32T^{2} \)
67 \( 1 - 1.62e16T + 7.40e32T^{2} \)
71 \( 1 - 3.19e16iT - 2.10e33T^{2} \)
73 \( 1 + 6.00e16T + 3.46e33T^{2} \)
79 \( 1 + 1.80e17T + 1.43e34T^{2} \)
83 \( 1 + 8.49e16iT - 3.49e34T^{2} \)
89 \( 1 + 5.84e17iT - 1.22e35T^{2} \)
97 \( 1 + 1.70e17T + 5.77e35T^{2} \)
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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

−18.81165512256499529327187625592, −17.09706261204694073565059913925, −16.11054790008726661032457360954, −12.93698323774024478049552779189, −12.56126893844413146007477848754, −10.57751384456839864342271764464, −8.808047610760058026605869492421, −6.04050552282655351767962832158, −4.24984511827496006736782550584, −1.36447798711501239219946032905, 0.00306737089505830738251974423, 3.51252611718057511956965275021, 6.00557962546592460698305099099, 6.95601463872881762348848010645, 9.793862659824189282330190637081, 11.24425454621489668856055647558, 13.41706102605803721078177204396, 15.28002393663555026796355039246, 16.34037577570337190945402210503, 18.00726760197210690296629671897

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