Properties

Degree 2
Conductor 3
Sign $-0.998 + 0.0628i$
Motivic weight 38
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.84e5i·2-s + (1.15e9 − 7.30e7i)3-s − 3.40e11·4-s + 3.07e13i·5-s + (5.72e13 + 9.09e14i)6-s + 1.07e16·7-s − 5.15e16i·8-s + (1.34e18 − 1.69e17i)9-s − 2.41e19·10-s + 2.63e19i·11-s + (−3.95e20 + 2.48e19i)12-s + 2.34e21·13-s + 8.44e21i·14-s + (2.24e21 + 3.56e22i)15-s − 5.31e22·16-s + 1.67e23i·17-s + ⋯
L(s)  = 1  + 1.49i·2-s + (0.998 − 0.0628i)3-s − 1.23·4-s + 1.61i·5-s + (0.0939 + 1.49i)6-s + 0.944·7-s − 0.357i·8-s + (0.992 − 0.125i)9-s − 2.41·10-s + 0.431i·11-s + (−1.23 + 0.0778i)12-s + 1.60·13-s + 1.41i·14-s + (0.101 + 1.60i)15-s − 0.704·16-s + 0.699i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0628i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (-0.998 + 0.0628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $-0.998 + 0.0628i$
motivic weight  =  \(38\)
character  :  $\chi_{3} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3,\ (\ :19),\ -0.998 + 0.0628i)\)
\(L(\frac{39}{2})\)  \(\approx\)  \(3.172611920\)
\(L(\frac12)\)  \(\approx\)  \(3.172611920\)
\(L(20)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3$,\(F_p(T)\) is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-1.15e9 + 7.30e7i)T \)
good2 \( 1 - 7.84e5iT - 2.74e11T^{2} \)
5 \( 1 - 3.07e13iT - 3.63e26T^{2} \)
7 \( 1 - 1.07e16T + 1.29e32T^{2} \)
11 \( 1 - 2.63e19iT - 3.74e39T^{2} \)
13 \( 1 - 2.34e21T + 2.13e42T^{2} \)
17 \( 1 - 1.67e23iT - 5.71e46T^{2} \)
19 \( 1 + 9.52e23T + 3.91e48T^{2} \)
23 \( 1 + 4.84e25iT - 5.56e51T^{2} \)
29 \( 1 + 8.06e27iT - 3.72e55T^{2} \)
31 \( 1 + 1.15e28T + 4.69e56T^{2} \)
37 \( 1 + 4.26e29T + 3.90e59T^{2} \)
41 \( 1 + 7.27e30iT - 1.93e61T^{2} \)
43 \( 1 - 9.82e30T + 1.17e62T^{2} \)
47 \( 1 - 8.44e29iT - 3.46e63T^{2} \)
53 \( 1 + 6.22e31iT - 3.33e65T^{2} \)
59 \( 1 - 5.85e33iT - 1.96e67T^{2} \)
61 \( 1 + 2.26e33T + 6.95e67T^{2} \)
67 \( 1 + 2.71e34T + 2.45e69T^{2} \)
71 \( 1 - 6.76e34iT - 2.22e70T^{2} \)
73 \( 1 - 4.44e35T + 6.40e70T^{2} \)
79 \( 1 + 1.68e35T + 1.28e72T^{2} \)
83 \( 1 + 3.55e36iT - 8.41e72T^{2} \)
89 \( 1 - 7.51e36iT - 1.19e74T^{2} \)
97 \( 1 - 2.81e37T + 3.14e75T^{2} \)
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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

−17.75018207290343482842479867924, −15.52627310338877139320060818037, −14.71791855984913760111674856798, −13.79261208757040209536175164664, −10.75330759756829121384180538349, −8.549551885025680265123160091505, −7.43338794391056291355562397782, −6.23619079578136986090723537076, −3.96814825645353378177652618033, −2.11971244430886432161087341781, 1.01850926545568037647024150277, 1.70241565015694465514505831689, 3.52380295569413181016982407799, 4.77509656543402026298813144505, 8.352658978428228735761060788413, 9.229459125601500329545101451532, 11.12329227227169041499068487275, 12.71137597707358326054397843564, 13.75269438269912781823248165509, 16.04450937792367148868761198581

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