Properties

Degree 2
Conductor $ 2 \cdot 3 $
Sign $-0.135 - 0.990i$
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 + (1.95e4 − 2.66e3i)3-s − 1.31e5·4-s + 1.30e6i·5-s + (9.63e5 + 7.06e6i)6-s + 3.15e7·7-s − 4.74e7i·8-s + (3.73e8 − 1.03e8i)9-s − 4.72e8·10-s + 3.81e9i·11-s + (−2.55e9 + 3.48e8i)12-s − 5.50e9·13-s + 1.14e10i·14-s + (3.47e9 + 2.54e10i)15-s + 1.71e10·16-s + 1.95e11i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.990 − 0.135i)3-s − 0.500·4-s + 0.668i·5-s + (0.0955 + 0.700i)6-s + 0.782·7-s − 0.353i·8-s + (0.963 − 0.267i)9-s − 0.472·10-s + 1.61i·11-s + (−0.495 + 0.0675i)12-s − 0.519·13-s + 0.553i·14-s + (0.0903 + 0.662i)15-s + 0.250·16-s + 1.65i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(19-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6\)    =    \(2 \cdot 3\)
\( \varepsilon \)  =  $-0.135 - 0.990i$
motivic weight  =  \(18\)
character  :  $\chi_{6} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6,\ (\ :9),\ -0.135 - 0.990i)$
$L(\frac{19}{2})$  $\approx$  $1.60306 + 1.83661i$
$L(\frac12)$  $\approx$  $1.60306 + 1.83661i$
$L(10)$   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 \( 1 + (-1.95e4 + 2.66e3i)T \)
good5 \( 1 - 1.30e6iT - 3.81e12T^{2} \)
7 \( 1 - 3.15e7T + 1.62e15T^{2} \)
11 \( 1 - 3.81e9iT - 5.55e18T^{2} \)
13 \( 1 + 5.50e9T + 1.12e20T^{2} \)
17 \( 1 - 1.95e11iT - 1.40e22T^{2} \)
19 \( 1 + 1.56e11T + 1.04e23T^{2} \)
23 \( 1 + 1.49e12iT - 3.24e24T^{2} \)
29 \( 1 + 7.21e12iT - 2.10e26T^{2} \)
31 \( 1 - 3.81e12T + 6.99e26T^{2} \)
37 \( 1 - 2.50e14T + 1.68e28T^{2} \)
41 \( 1 + 4.39e14iT - 1.07e29T^{2} \)
43 \( 1 + 5.41e14T + 2.52e29T^{2} \)
47 \( 1 + 1.94e15iT - 1.25e30T^{2} \)
53 \( 1 - 1.53e15iT - 1.08e31T^{2} \)
59 \( 1 + 6.50e15iT - 7.50e31T^{2} \)
61 \( 1 - 3.87e15T + 1.36e32T^{2} \)
67 \( 1 - 3.71e16T + 7.40e32T^{2} \)
71 \( 1 - 6.33e16iT - 2.10e33T^{2} \)
73 \( 1 + 5.51e16T + 3.46e33T^{2} \)
79 \( 1 + 7.16e16T + 1.43e34T^{2} \)
83 \( 1 + 1.18e17iT - 3.49e34T^{2} \)
89 \( 1 + 9.49e16iT - 1.22e35T^{2} \)
97 \( 1 + 3.38e17T + 5.77e35T^{2} \)
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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

−18.65448760198895053141335072508, −17.37079024628994687358664936901, −15.03235516249335234199423719748, −14.64300984658367303239727490826, −12.74796209308551094788426831219, −10.11560295658671357009838399503, −8.256047643991086991922305836148, −6.92455988168256863984314958708, −4.33019921875572607085754437859, −2.06752858939011897345490473136, 1.06296280784780493999662929962, 2.91397138753411590374262218766, 4.78617867487877891476828947179, 8.098142462528363529437747176132, 9.355958781691056624680816120701, 11.34434913392810402102856637120, 13.22830772240933721270975886282, 14.42333739638818187133899590036, 16.37460657873973894258115135750, 18.38647295708425014017172090512

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