Properties

Degree 2
Conductor 3
Sign $0.662 + 0.749i$
Motivic weight 16
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 198. i·2-s + (4.34e3 + 4.91e3i)3-s + 2.60e4·4-s − 4.82e5i·5-s + (9.77e5 − 8.63e5i)6-s + 5.53e6·7-s − 1.81e7i·8-s + (−5.31e6 + 4.27e7i)9-s − 9.58e7·10-s + 1.03e8i·11-s + (1.13e8 + 1.27e8i)12-s − 7.54e7·13-s − 1.10e9i·14-s + (2.37e9 − 2.09e9i)15-s − 1.91e9·16-s + 7.90e9i·17-s + ⋯
L(s)  = 1  − 0.776i·2-s + (0.662 + 0.749i)3-s + 0.397·4-s − 1.23i·5-s + (0.581 − 0.514i)6-s + 0.960·7-s − 1.08i·8-s + (−0.123 + 0.992i)9-s − 0.958·10-s + 0.484i·11-s + (0.262 + 0.297i)12-s − 0.0925·13-s − 0.745i·14-s + (0.925 − 0.817i)15-s − 0.445·16-s + 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(17-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $0.662 + 0.749i$
motivic weight  =  \(16\)
character  :  $\chi_{3} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3,\ (\ :8),\ 0.662 + 0.749i)$
$L(\frac{17}{2})$  $\approx$  $1.96194 - 0.884693i$
$L(\frac12)$  $\approx$  $1.96194 - 0.884693i$
$L(9)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + (-4.34e3 - 4.91e3i)T \)
good2 \( 1 + 198. iT - 6.55e4T^{2} \)
5 \( 1 + 4.82e5iT - 1.52e11T^{2} \)
7 \( 1 - 5.53e6T + 3.32e13T^{2} \)
11 \( 1 - 1.03e8iT - 4.59e16T^{2} \)
13 \( 1 + 7.54e7T + 6.65e17T^{2} \)
17 \( 1 - 7.90e9iT - 4.86e19T^{2} \)
19 \( 1 + 3.27e10T + 2.88e20T^{2} \)
23 \( 1 + 4.02e10iT - 6.13e21T^{2} \)
29 \( 1 - 6.12e11iT - 2.50e23T^{2} \)
31 \( 1 - 5.76e11T + 7.27e23T^{2} \)
37 \( 1 + 2.03e11T + 1.23e25T^{2} \)
41 \( 1 + 2.51e12iT - 6.37e25T^{2} \)
43 \( 1 - 2.81e12T + 1.36e26T^{2} \)
47 \( 1 + 1.80e13iT - 5.66e26T^{2} \)
53 \( 1 - 2.59e13iT - 3.87e27T^{2} \)
59 \( 1 - 1.63e14iT - 2.15e28T^{2} \)
61 \( 1 - 7.04e13T + 3.67e28T^{2} \)
67 \( 1 + 1.75e14T + 1.64e29T^{2} \)
71 \( 1 + 6.72e14iT - 4.16e29T^{2} \)
73 \( 1 - 1.04e15T + 6.50e29T^{2} \)
79 \( 1 - 4.55e13T + 2.30e30T^{2} \)
83 \( 1 - 3.65e14iT - 5.07e30T^{2} \)
89 \( 1 + 6.68e15iT - 1.54e31T^{2} \)
97 \( 1 + 5.40e15T + 6.14e31T^{2} \)
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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

−21.47004427375102679727845449304, −20.70679434638244207497097708227, −19.58068433352889178502111689657, −16.79486210952798485715116714471, −15.04846567527132458154960033715, −12.64276244970839965742851531576, −10.59660824468351979174914713129, −8.558340964066520449583168421918, −4.40448525074930780880881606505, −1.83488451980409632271054600575, 2.41699071672461749929038480110, 6.56313577358796467533377958539, 8.000104043634496999618136882987, 11.31120669693049391981952110807, 14.12165222353993348217533209508, 15.14068280125796332056697862769, 17.58366445996388261272257605868, 19.09383622882766743650791043802, 20.96792391894045932482942485181, 23.31168724302517195449186022008

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