Properties

Degree 2
Conductor 43
Sign $0.0894 - 0.995i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.186 − 0.234i)2-s + (−17.0 − 21.4i)3-s + (7.10 + 31.1i)4-s + (−36.3 − 17.4i)5-s − 8.19·6-s + 50.2·7-s + (17.2 + 8.30i)8-s + (−112. + 493. i)9-s + (−10.8 + 5.23i)10-s + (−100. + 441. i)11-s + (544. − 682. i)12-s + (894. + 430. i)13-s + (9.38 − 11.7i)14-s + (245. + 1.07e3i)15-s + (−914. + 440. i)16-s + (−1.43e3 + 688. i)17-s + ⋯
L(s)  = 1  + (0.0329 − 0.0413i)2-s + (−1.09 − 1.37i)3-s + (0.221 + 0.972i)4-s + (−0.649 − 0.312i)5-s − 0.0929·6-s + 0.387·7-s + (0.0952 + 0.0458i)8-s + (−0.463 + 2.03i)9-s + (−0.0343 + 0.0165i)10-s + (−0.250 + 1.09i)11-s + (1.09 − 1.36i)12-s + (1.46 + 0.707i)13-s + (0.0127 − 0.0160i)14-s + (0.281 + 1.23i)15-s + (−0.893 + 0.430i)16-s + (−1.20 + 0.578i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0894 - 0.995i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0894 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.0894 - 0.995i$
motivic weight  =  \(5\)
character  :  $\chi_{43} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ 0.0894 - 0.995i)\)
\(L(3)\)  \(\approx\)  \(0.442367 + 0.404426i\)
\(L(\frac12)\)  \(\approx\)  \(0.442367 + 0.404426i\)
\(L(\frac{7}{2})\)   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 \( 1 + (2.89e3 - 1.17e4i)T \)
good2 \( 1 + (-0.186 + 0.234i)T + (-7.12 - 31.1i)T^{2} \)
3 \( 1 + (17.0 + 21.4i)T + (-54.0 + 236. i)T^{2} \)
5 \( 1 + (36.3 + 17.4i)T + (1.94e3 + 2.44e3i)T^{2} \)
7 \( 1 - 50.2T + 1.68e4T^{2} \)
11 \( 1 + (100. - 441. i)T + (-1.45e5 - 6.98e4i)T^{2} \)
13 \( 1 + (-894. - 430. i)T + (2.31e5 + 2.90e5i)T^{2} \)
17 \( 1 + (1.43e3 - 688. i)T + (8.85e5 - 1.11e6i)T^{2} \)
19 \( 1 + (308. + 1.35e3i)T + (-2.23e6 + 1.07e6i)T^{2} \)
23 \( 1 + (512. - 2.24e3i)T + (-5.79e6 - 2.79e6i)T^{2} \)
29 \( 1 + (3.69e3 - 4.63e3i)T + (-4.56e6 - 1.99e7i)T^{2} \)
31 \( 1 + (-3.75e3 + 4.71e3i)T + (-6.37e6 - 2.79e7i)T^{2} \)
37 \( 1 + 729.T + 6.93e7T^{2} \)
41 \( 1 + (6.48e3 - 8.12e3i)T + (-2.57e7 - 1.12e8i)T^{2} \)
47 \( 1 + (-2.61e3 - 1.14e4i)T + (-2.06e8 + 9.95e7i)T^{2} \)
53 \( 1 + (-1.69e4 + 8.17e3i)T + (2.60e8 - 3.26e8i)T^{2} \)
59 \( 1 + (-4.06e3 + 1.95e3i)T + (4.45e8 - 5.58e8i)T^{2} \)
61 \( 1 + (-1.02e4 - 1.28e4i)T + (-1.87e8 + 8.23e8i)T^{2} \)
67 \( 1 + (-2.93e3 - 1.28e4i)T + (-1.21e9 + 5.85e8i)T^{2} \)
71 \( 1 + (-4.20e3 - 1.84e4i)T + (-1.62e9 + 7.82e8i)T^{2} \)
73 \( 1 + (5.74e4 + 2.76e4i)T + (1.29e9 + 1.62e9i)T^{2} \)
79 \( 1 - 7.07e4T + 3.07e9T^{2} \)
83 \( 1 + (1.99e4 + 2.50e4i)T + (-8.76e8 + 3.84e9i)T^{2} \)
89 \( 1 + (7.77e4 + 9.74e4i)T + (-1.24e9 + 5.44e9i)T^{2} \)
97 \( 1 + (1.42e4 - 6.22e4i)T + (-7.73e9 - 3.72e9i)T^{2} \)
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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

−15.54815090586984934760753826451, −13.37247538272264664794817529736, −12.83211566037509933982948539038, −11.61662636140262267279605355085, −11.22919478316267941716540324743, −8.506615195883587488603807798625, −7.44640047005053016705140365809, −6.43542193458248673190907358167, −4.43148098851156780172171966682, −1.78747508427080132587383124710, 0.36773611873695580275510870842, 3.85389052213728288408756222457, 5.33834589586572706044369352187, 6.31826143458739582572486907602, 8.656861988837320015736164408889, 10.28338219068792215445719675157, 10.97179559208865440930893709982, 11.59161059214770336764294406100, 13.77197665885488525254719185571, 15.24352196918713218716942106934

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