Properties

Degree 2
Conductor 3
Sign $-0.247 + 0.968i$
Motivic weight 38
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.40e5i·2-s + (2.88e8 − 1.12e9i)3-s − 1.34e11·4-s − 3.15e13i·5-s + (7.20e14 + 1.84e14i)6-s + 9.50e15·7-s + 8.96e16i·8-s + (−1.18e18 − 6.48e17i)9-s + 2.02e19·10-s − 6.59e19i·11-s + (−3.88e19 + 1.51e20i)12-s − 1.55e21·13-s + 6.08e21i·14-s + (−3.55e22 − 9.10e21i)15-s − 9.44e22·16-s + 2.67e23i·17-s + ⋯
L(s)  = 1  + 1.22i·2-s + (0.247 − 0.968i)3-s − 0.490·4-s − 1.65i·5-s + (1.18 + 0.302i)6-s + 0.833·7-s + 0.622i·8-s + (−0.877 − 0.480i)9-s + 2.02·10-s − 1.07i·11-s + (−0.121 + 0.475i)12-s − 1.06·13-s + 1.01i·14-s + (−1.60 − 0.410i)15-s − 1.24·16-s + 1.11i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $-0.247 + 0.968i$
motivic weight  =  \(38\)
character  :  $\chi_{3} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3,\ (\ :19),\ -0.247 + 0.968i)\)
\(L(\frac{39}{2})\)  \(\approx\)  \(1.483586902\)
\(L(\frac12)\)  \(\approx\)  \(1.483586902\)
\(L(20)\)   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 \( 1 + (-2.88e8 + 1.12e9i)T \)
good2 \( 1 - 6.40e5iT - 2.74e11T^{2} \)
5 \( 1 + 3.15e13iT - 3.63e26T^{2} \)
7 \( 1 - 9.50e15T + 1.29e32T^{2} \)
11 \( 1 + 6.59e19iT - 3.74e39T^{2} \)
13 \( 1 + 1.55e21T + 2.13e42T^{2} \)
17 \( 1 - 2.67e23iT - 5.71e46T^{2} \)
19 \( 1 + 1.27e24T + 3.91e48T^{2} \)
23 \( 1 + 4.27e25iT - 5.56e51T^{2} \)
29 \( 1 + 6.33e27iT - 3.72e55T^{2} \)
31 \( 1 - 2.54e28T + 4.69e56T^{2} \)
37 \( 1 + 9.87e28T + 3.90e59T^{2} \)
41 \( 1 - 3.57e30iT - 1.93e61T^{2} \)
43 \( 1 + 8.11e30T + 1.17e62T^{2} \)
47 \( 1 + 7.20e31iT - 3.46e63T^{2} \)
53 \( 1 + 4.09e32iT - 3.33e65T^{2} \)
59 \( 1 + 1.03e33iT - 1.96e67T^{2} \)
61 \( 1 - 5.64e33T + 6.95e67T^{2} \)
67 \( 1 + 6.14e34T + 2.45e69T^{2} \)
71 \( 1 + 1.97e35iT - 2.22e70T^{2} \)
73 \( 1 + 1.80e35T + 6.40e70T^{2} \)
79 \( 1 - 1.59e36T + 1.28e72T^{2} \)
83 \( 1 + 2.62e36iT - 8.41e72T^{2} \)
89 \( 1 - 3.89e36iT - 1.19e74T^{2} \)
97 \( 1 - 8.00e37T + 3.14e75T^{2} \)
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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

−16.69035106440726561167192259903, −14.83350921494843462020516035586, −13.37176869796382613724606233071, −11.86852717345284165841583416478, −8.559262548029472542114230858174, −8.030193690838704753445957122330, −6.15285707376197621117857449492, −4.84529229722841597485721194836, −1.90772571939114440967485893683, −0.43003190831744765631227611582, 2.15414724657371109330246305388, 3.04062994775100672200881936735, 4.58773088106365703037438730344, 7.19622887609242982210602887747, 9.732004603716303079123789328340, 10.66527986822830146886156624226, 11.75673618106568217883730367259, 14.26799497044820584635550176947, 15.32265374095719305066437116919, 17.78562051539869486744372795428

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