Properties

Degree 2
Conductor 43
Sign $-0.968 - 0.250i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.45 − 8.10i)2-s + (−16.9 − 21.2i)3-s + (−16.7 − 73.4i)4-s + (36.1 + 17.4i)5-s − 281.·6-s + 237.·7-s + (−404. − 194. i)8-s + (−110. + 482. i)9-s + (374. − 180. i)10-s + (−32.8 + 144. i)11-s + (−1.27e3 + 1.59e3i)12-s + (−470. − 226. i)13-s + (1.53e3 − 1.92e3i)14-s + (−242. − 1.06e3i)15-s + (−2.01e3 + 972. i)16-s + (1.13e3 − 547. i)17-s + ⋯
L(s)  = 1  + (1.14 − 1.43i)2-s + (−1.08 − 1.36i)3-s + (−0.523 − 2.29i)4-s + (0.646 + 0.311i)5-s − 3.19·6-s + 1.83·7-s + (−2.23 − 1.07i)8-s + (−0.453 + 1.98i)9-s + (1.18 − 0.570i)10-s + (−0.0819 + 0.359i)11-s + (−2.55 + 3.20i)12-s + (−0.771 − 0.371i)13-s + (2.09 − 2.62i)14-s + (−0.278 − 1.21i)15-s + (−1.97 + 0.949i)16-s + (0.953 − 0.459i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.968 - 0.250i$
motivic weight  =  \(5\)
character  :  $\chi_{43} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ -0.968 - 0.250i)\)
\(L(3)\)  \(\approx\)  \(0.302490 + 2.37721i\)
\(L(\frac12)\)  \(\approx\)  \(0.302490 + 2.37721i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-1.11e4 - 4.77e3i)T \)
good2 \( 1 + (-6.45 + 8.10i)T + (-7.12 - 31.1i)T^{2} \)
3 \( 1 + (16.9 + 21.2i)T + (-54.0 + 236. i)T^{2} \)
5 \( 1 + (-36.1 - 17.4i)T + (1.94e3 + 2.44e3i)T^{2} \)
7 \( 1 - 237.T + 1.68e4T^{2} \)
11 \( 1 + (32.8 - 144. i)T + (-1.45e5 - 6.98e4i)T^{2} \)
13 \( 1 + (470. + 226. i)T + (2.31e5 + 2.90e5i)T^{2} \)
17 \( 1 + (-1.13e3 + 547. i)T + (8.85e5 - 1.11e6i)T^{2} \)
19 \( 1 + (-347. - 1.52e3i)T + (-2.23e6 + 1.07e6i)T^{2} \)
23 \( 1 + (-280. + 1.23e3i)T + (-5.79e6 - 2.79e6i)T^{2} \)
29 \( 1 + (2.61e3 - 3.27e3i)T + (-4.56e6 - 1.99e7i)T^{2} \)
31 \( 1 + (2.74e3 - 3.44e3i)T + (-6.37e6 - 2.79e7i)T^{2} \)
37 \( 1 - 4.49e3T + 6.93e7T^{2} \)
41 \( 1 + (122. - 153. i)T + (-2.57e7 - 1.12e8i)T^{2} \)
47 \( 1 + (652. + 2.85e3i)T + (-2.06e8 + 9.95e7i)T^{2} \)
53 \( 1 + (-1.04e4 + 5.00e3i)T + (2.60e8 - 3.26e8i)T^{2} \)
59 \( 1 + (1.58e4 - 7.61e3i)T + (4.45e8 - 5.58e8i)T^{2} \)
61 \( 1 + (2.66e4 + 3.34e4i)T + (-1.87e8 + 8.23e8i)T^{2} \)
67 \( 1 + (-5.80e3 - 2.54e4i)T + (-1.21e9 + 5.85e8i)T^{2} \)
71 \( 1 + (-8.66e3 - 3.79e4i)T + (-1.62e9 + 7.82e8i)T^{2} \)
73 \( 1 + (-3.33e4 - 1.60e4i)T + (1.29e9 + 1.62e9i)T^{2} \)
79 \( 1 - 6.36e3T + 3.07e9T^{2} \)
83 \( 1 + (4.72e4 + 5.92e4i)T + (-8.76e8 + 3.84e9i)T^{2} \)
89 \( 1 + (3.88e3 + 4.87e3i)T + (-1.24e9 + 5.44e9i)T^{2} \)
97 \( 1 + (-2.56e4 + 1.12e5i)T + (-7.73e9 - 3.72e9i)T^{2} \)
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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

−14.10180869858372678390723310799, −12.73486082309480582145496877993, −12.04919422626006815980974863290, −11.19500782109128614012111726695, −10.24570764252045268967577043559, −7.58662462748948031593381980761, −5.74737147123389624057956927220, −4.95424630101547833884522927331, −2.18811045947755684331037022052, −1.21665265369675820918986172346, 4.21115957993382679765796496161, 5.16322890670570577843950089654, 5.76331441580933046701878454429, 7.67690270591312450050877552860, 9.294576608328385760319489538975, 11.07935103056246929878669948239, 12.06618929417731679992700113442, 13.72217055000979176132558522692, 14.78135028271277725933674841965, 15.36542920687339861944932116268

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