Properties

Degree 2
Conductor 43
Sign $-0.0329 - 0.999i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.61 + 5.42i)2-s + (1.01 − 2.11i)3-s + (17.2 − 21.6i)4-s + (−102. − 23.4i)5-s + 14.1·6-s + 486. i·7-s + (538. + 122. i)8-s + (451. + 565. i)9-s + (−141. − 619. i)10-s + (609. + 763. i)11-s + (−28.1 − 58.4i)12-s + (−77.4 + 339. i)13-s + (−2.63e3 + 1.27e3i)14-s + (−154. + 193. i)15-s + (346. + 1.51e3i)16-s + (664. + 2.91e3i)17-s + ⋯
L(s)  = 1  + (0.326 + 0.678i)2-s + (0.0376 − 0.0781i)3-s + (0.269 − 0.338i)4-s + (−0.822 − 0.187i)5-s + 0.0653·6-s + 1.41i·7-s + (1.05 + 0.240i)8-s + (0.618 + 0.775i)9-s + (−0.141 − 0.619i)10-s + (0.457 + 0.573i)11-s + (−0.0162 − 0.0338i)12-s + (−0.0352 + 0.154i)13-s + (−0.961 + 0.463i)14-s + (−0.0456 + 0.0572i)15-s + (0.0845 + 0.370i)16-s + (0.135 + 0.592i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0329 - 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0329 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.0329 - 0.999i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.0329 - 0.999i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(1.42987 + 1.47772i\)
\(L(\frac12)\)  \(\approx\)  \(1.42987 + 1.47772i\)
\(L(4)\)   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 \( 1 + (4.22e4 + 6.73e4i)T \)
good2 \( 1 + (-2.61 - 5.42i)T + (-39.9 + 50.0i)T^{2} \)
3 \( 1 + (-1.01 + 2.11i)T + (-454. - 569. i)T^{2} \)
5 \( 1 + (102. + 23.4i)T + (1.40e4 + 6.77e3i)T^{2} \)
7 \( 1 - 486. iT - 1.17e5T^{2} \)
11 \( 1 + (-609. - 763. i)T + (-3.94e5 + 1.72e6i)T^{2} \)
13 \( 1 + (77.4 - 339. i)T + (-4.34e6 - 2.09e6i)T^{2} \)
17 \( 1 + (-664. - 2.91e3i)T + (-2.17e7 + 1.04e7i)T^{2} \)
19 \( 1 + (-2.01e3 - 1.60e3i)T + (1.04e7 + 4.58e7i)T^{2} \)
23 \( 1 + (5.36e3 + 6.73e3i)T + (-3.29e7 + 1.44e8i)T^{2} \)
29 \( 1 + (-1.62e3 - 3.37e3i)T + (-3.70e8 + 4.65e8i)T^{2} \)
31 \( 1 + (-4.57e3 + 2.20e3i)T + (5.53e8 - 6.93e8i)T^{2} \)
37 \( 1 - 1.45e4iT - 2.56e9T^{2} \)
41 \( 1 + (-3.40e4 + 1.64e4i)T + (2.96e9 - 3.71e9i)T^{2} \)
47 \( 1 + (-4.89e4 + 6.14e4i)T + (-2.39e9 - 1.05e10i)T^{2} \)
53 \( 1 + (-2.81e4 - 1.23e5i)T + (-1.99e10 + 9.61e9i)T^{2} \)
59 \( 1 + (-1.65e4 - 7.24e4i)T + (-3.80e10 + 1.83e10i)T^{2} \)
61 \( 1 + (7.61e3 - 1.58e4i)T + (-3.21e10 - 4.02e10i)T^{2} \)
67 \( 1 + (-2.13e5 + 2.68e5i)T + (-2.01e10 - 8.81e10i)T^{2} \)
71 \( 1 + (-3.93e5 - 3.13e5i)T + (2.85e10 + 1.24e11i)T^{2} \)
73 \( 1 + (5.21e5 + 1.19e5i)T + (1.36e11 + 6.56e10i)T^{2} \)
79 \( 1 + 1.97e5T + 2.43e11T^{2} \)
83 \( 1 + (8.33e5 + 4.01e5i)T + (2.03e11 + 2.55e11i)T^{2} \)
89 \( 1 + (-5.03e5 + 1.04e6i)T + (-3.09e11 - 3.88e11i)T^{2} \)
97 \( 1 + (9.83e5 + 1.23e6i)T + (-1.85e11 + 8.12e11i)T^{2} \)
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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

−15.27303745872382120642437250920, −14.13829095885952109328437413398, −12.60064717208094598895783974583, −11.62869600202483285560588289275, −10.14468120275454362303532485650, −8.454924038670309752779909980767, −7.26763735069729209961641946701, −5.82081972715839534067117175475, −4.47007942866053697438356991291, −1.95011903278971210143197655848, 0.951816888768605929354005447606, 3.40728619316647817324425028805, 4.19998022739354058346380511652, 6.89688532098739444078007082864, 7.80129729849022143819071508449, 9.820794117450541012096366797573, 11.06764285222671887789495935864, 11.86109881073683616049276019336, 13.10291338934325604285202722143, 14.14633877426003995920455438856

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