Properties

Degree 2
Conductor 43
Sign $0.648 + 0.761i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.66 − 5.54i)2-s + (−33.1 − 2.48i)3-s + (16.2 + 20.4i)4-s + (−16.0 + 51.8i)5-s + (−102. + 177. i)6-s + (211. − 121. i)7-s + (540. − 123. i)8-s + (370. + 55.8i)9-s + (244. + 227. i)10-s + (250. − 314. i)11-s + (−489. − 717. i)12-s + (1.65e3 − 1.53e3i)13-s + (−112. − 1.49e3i)14-s + (659. − 1.67e3i)15-s + (387. − 1.69e3i)16-s + (5.16e3 − 1.59e3i)17-s + ⋯
L(s)  = 1  + (0.333 − 0.692i)2-s + (−1.22 − 0.0919i)3-s + (0.254 + 0.319i)4-s + (−0.128 + 0.415i)5-s + (−0.473 + 0.819i)6-s + (0.615 − 0.355i)7-s + (1.05 − 0.241i)8-s + (0.508 + 0.0766i)9-s + (0.244 + 0.227i)10-s + (0.188 − 0.236i)11-s + (−0.283 − 0.415i)12-s + (0.751 − 0.697i)13-s + (−0.0408 − 0.545i)14-s + (0.195 − 0.497i)15-s + (0.0945 − 0.414i)16-s + (1.05 − 0.324i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.648 + 0.761i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.648 + 0.761i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(1.52800 - 0.706056i\)
\(L(\frac12)\)  \(\approx\)  \(1.52800 - 0.706056i\)
\(L(4)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 \( 1 + (4.48e4 - 6.56e4i)T \)
good2 \( 1 + (-2.66 + 5.54i)T + (-39.9 - 50.0i)T^{2} \)
3 \( 1 + (33.1 + 2.48i)T + (720. + 108. i)T^{2} \)
5 \( 1 + (16.0 - 51.8i)T + (-1.29e4 - 8.80e3i)T^{2} \)
7 \( 1 + (-211. + 121. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (-250. + 314. i)T + (-3.94e5 - 1.72e6i)T^{2} \)
13 \( 1 + (-1.65e3 + 1.53e3i)T + (3.60e5 - 4.81e6i)T^{2} \)
17 \( 1 + (-5.16e3 + 1.59e3i)T + (1.99e7 - 1.35e7i)T^{2} \)
19 \( 1 + (169. + 1.12e3i)T + (-4.49e7 + 1.38e7i)T^{2} \)
23 \( 1 + (-72.1 - 183. i)T + (-1.08e8 + 1.00e8i)T^{2} \)
29 \( 1 + (-2.07e4 + 1.55e3i)T + (5.88e8 - 8.86e7i)T^{2} \)
31 \( 1 + (-2.53e3 + 1.73e3i)T + (3.24e8 - 8.26e8i)T^{2} \)
37 \( 1 + (3.92e3 + 2.26e3i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (4.25e4 + 2.05e4i)T + (2.96e9 + 3.71e9i)T^{2} \)
47 \( 1 + (3.58e4 + 4.49e4i)T + (-2.39e9 + 1.05e10i)T^{2} \)
53 \( 1 + (8.91e4 + 8.26e4i)T + (1.65e9 + 2.21e10i)T^{2} \)
59 \( 1 + (-3.27e4 + 1.43e5i)T + (-3.80e10 - 1.83e10i)T^{2} \)
61 \( 1 + (-1.18e5 + 1.73e5i)T + (-1.88e10 - 4.79e10i)T^{2} \)
67 \( 1 + (-1.44e5 + 2.17e4i)T + (8.64e10 - 2.66e10i)T^{2} \)
71 \( 1 + (4.66e5 + 1.82e5i)T + (9.39e10 + 8.71e10i)T^{2} \)
73 \( 1 + (-3.62e5 - 3.90e5i)T + (-1.13e10 + 1.50e11i)T^{2} \)
79 \( 1 + (3.15e3 + 5.47e3i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-4.48e4 + 5.98e5i)T + (-3.23e11 - 4.87e10i)T^{2} \)
89 \( 1 + (-5.60e5 - 4.20e4i)T + (4.91e11 + 7.40e10i)T^{2} \)
97 \( 1 + (5.49e5 - 6.88e5i)T + (-1.85e11 - 8.12e11i)T^{2} \)
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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

−14.32525374285749244960824530900, −12.99820484229392739358601237230, −11.87330748543355364864311181708, −11.15946880529031846917877692498, −10.37861828388599959747519724985, −8.005337941429999844685606831246, −6.61694577006521330015936456194, −5.04076248419831578915803218114, −3.30252089205355888084830335989, −1.07786089165903157524754899620, 1.29322560501966532821082125153, 4.64122411989143257422456988405, 5.64693261556772305336385307477, 6.70551677302985013509582799549, 8.361378931635665241328210195910, 10.29608188885372370654922711192, 11.37921869523210206389625120635, 12.27042328653882449379176513985, 13.94382805352679331875871489510, 14.98909888992481991410430226175

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