Properties

Degree 2
Conductor $ 2 \cdot 3 $
Sign $-0.974 + 0.225i$
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 + (1.47e3 + 6.39e3i)3-s − 3.27e4·4-s + 4.41e5i·5-s + (−1.15e6 + 2.67e5i)6-s + 2.81e6·7-s − 5.93e6i·8-s + (−3.86e7 + 1.89e7i)9-s − 7.99e7·10-s − 1.92e8i·11-s + (−4.84e7 − 2.09e8i)12-s − 1.24e9·13-s + 5.09e8i·14-s + (−2.82e9 + 6.52e8i)15-s + 1.07e9·16-s + 1.41e9i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.225 + 0.974i)3-s − 0.500·4-s + 1.13i·5-s + (−0.688 + 0.159i)6-s + 0.488·7-s − 0.353i·8-s + (−0.898 + 0.439i)9-s − 0.799·10-s − 0.900i·11-s + (−0.112 − 0.487i)12-s − 1.52·13-s + 0.345i·14-s + (−1.10 + 0.254i)15-s + 0.250·16-s + 0.203i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(17-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6\)    =    \(2 \cdot 3\)
\( \varepsilon \)  =  $-0.974 + 0.225i$
motivic weight  =  \(16\)
character  :  $\chi_{6} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6,\ (\ :8),\ -0.974 + 0.225i)$
$L(\frac{17}{2})$  $\approx$  $0.156209 - 1.36800i$
$L(\frac12)$  $\approx$  $0.156209 - 1.36800i$
$L(9)$   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 \( 1 + (-1.47e3 - 6.39e3i)T \)
good5 \( 1 - 4.41e5iT - 1.52e11T^{2} \)
7 \( 1 - 2.81e6T + 3.32e13T^{2} \)
11 \( 1 + 1.92e8iT - 4.59e16T^{2} \)
13 \( 1 + 1.24e9T + 6.65e17T^{2} \)
17 \( 1 - 1.41e9iT - 4.86e19T^{2} \)
19 \( 1 - 2.68e10T + 2.88e20T^{2} \)
23 \( 1 - 1.20e11iT - 6.13e21T^{2} \)
29 \( 1 - 6.43e11iT - 2.50e23T^{2} \)
31 \( 1 + 1.14e12T + 7.27e23T^{2} \)
37 \( 1 - 4.25e12T + 1.23e25T^{2} \)
41 \( 1 - 3.12e12iT - 6.37e25T^{2} \)
43 \( 1 - 4.47e12T + 1.36e26T^{2} \)
47 \( 1 - 1.17e13iT - 5.66e26T^{2} \)
53 \( 1 + 4.41e13iT - 3.87e27T^{2} \)
59 \( 1 - 1.42e14iT - 2.15e28T^{2} \)
61 \( 1 - 2.29e13T + 3.67e28T^{2} \)
67 \( 1 + 1.94e14T + 1.64e29T^{2} \)
71 \( 1 - 3.85e13iT - 4.16e29T^{2} \)
73 \( 1 - 1.42e15T + 6.50e29T^{2} \)
79 \( 1 - 1.03e15T + 2.30e30T^{2} \)
83 \( 1 - 1.75e15iT - 5.07e30T^{2} \)
89 \( 1 - 1.35e15iT - 1.54e31T^{2} \)
97 \( 1 - 5.20e15T + 6.14e31T^{2} \)
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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

−19.67521459193108088936719153519, −17.98701544484567237230394586584, −16.43106337154318646533854041761, −14.95079615997017308981672637501, −14.12091066904340128673569666163, −11.18113866479690287502689900392, −9.558638246872137629440879838835, −7.54196060801267272916236767524, −5.33064888596798662705154190176, −3.22223620215909675317754291266, 0.64693485180959382816645114787, 2.20009177237340446336979869047, 4.91430216290693060769655079944, 7.69620351503980472519863286355, 9.400277279728883639703331128777, 11.91894440373373700332648051237, 12.81200929265294586500622232871, 14.45337498163477961641867625307, 17.00566813792454392864351703109, 18.24889059782971425391450812057

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