Properties

Degree 2
Conductor 43
Sign $0.893 + 0.449i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 − 2.26i)2-s + (21.5 + 3.25i)3-s + (5.25 − 23.0i)4-s + (62.1 + 42.3i)5-s + (−31.6 − 54.7i)6-s + (1.33 − 2.30i)7-s + (−145. + 69.9i)8-s + (222. + 68.6i)9-s + (−16.3 − 217. i)10-s + (124. + 544. i)11-s + (188. − 479. i)12-s + (82.1 − 1.09e3i)13-s + (−7.63 + 1.15i)14-s + (1.20e3 + 1.11e3i)15-s + (−259. − 124. i)16-s + (522. − 356. i)17-s + ⋯
L(s)  = 1  + (−0.319 − 0.400i)2-s + (1.38 + 0.208i)3-s + (0.164 − 0.718i)4-s + (1.11 + 0.758i)5-s + (−0.358 − 0.621i)6-s + (0.0102 − 0.0177i)7-s + (−0.802 + 0.386i)8-s + (0.916 + 0.282i)9-s + (−0.0515 − 0.687i)10-s + (0.309 + 1.35i)11-s + (0.377 − 0.960i)12-s + (0.134 − 1.79i)13-s + (−0.0104 + 0.00156i)14-s + (1.38 + 1.28i)15-s + (−0.253 − 0.121i)16-s + (0.438 − 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.893 + 0.449i$
motivic weight  =  \(5\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ 0.893 + 0.449i)\)
\(L(3)\)  \(\approx\)  \(2.38413 - 0.565765i\)
\(L(\frac12)\)  \(\approx\)  \(2.38413 - 0.565765i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-4.68e3 + 1.11e4i)T \)
good2 \( 1 + (1.80 + 2.26i)T + (-7.12 + 31.1i)T^{2} \)
3 \( 1 + (-21.5 - 3.25i)T + (232. + 71.6i)T^{2} \)
5 \( 1 + (-62.1 - 42.3i)T + (1.14e3 + 2.90e3i)T^{2} \)
7 \( 1 + (-1.33 + 2.30i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-124. - 544. i)T + (-1.45e5 + 6.98e4i)T^{2} \)
13 \( 1 + (-82.1 + 1.09e3i)T + (-3.67e5 - 5.53e4i)T^{2} \)
17 \( 1 + (-522. + 356. i)T + (5.18e5 - 1.32e6i)T^{2} \)
19 \( 1 + (-22.1 + 6.84i)T + (2.04e6 - 1.39e6i)T^{2} \)
23 \( 1 + (-495. + 459. i)T + (4.80e5 - 6.41e6i)T^{2} \)
29 \( 1 + (8.13e3 - 1.22e3i)T + (1.95e7 - 6.04e6i)T^{2} \)
31 \( 1 + (2.82e3 - 7.19e3i)T + (-2.09e7 - 1.94e7i)T^{2} \)
37 \( 1 + (2.44e3 + 4.24e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + (634. + 795. i)T + (-2.57e7 + 1.12e8i)T^{2} \)
47 \( 1 + (2.52e3 - 1.10e4i)T + (-2.06e8 - 9.95e7i)T^{2} \)
53 \( 1 + (465. + 6.21e3i)T + (-4.13e8 + 6.23e7i)T^{2} \)
59 \( 1 + (1.11e4 + 5.34e3i)T + (4.45e8 + 5.58e8i)T^{2} \)
61 \( 1 + (2.37e3 + 6.04e3i)T + (-6.19e8 + 5.74e8i)T^{2} \)
67 \( 1 + (6.57e4 - 2.02e4i)T + (1.11e9 - 7.60e8i)T^{2} \)
71 \( 1 + (-1.43e4 - 1.33e4i)T + (1.34e8 + 1.79e9i)T^{2} \)
73 \( 1 + (4.92e3 - 6.56e4i)T + (-2.04e9 - 3.08e8i)T^{2} \)
79 \( 1 + (-1.10e4 + 1.91e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-6.70e4 - 1.01e4i)T + (3.76e9 + 1.16e9i)T^{2} \)
89 \( 1 + (-6.76e4 - 1.01e4i)T + (5.33e9 + 1.64e9i)T^{2} \)
97 \( 1 + (1.41e4 + 6.18e4i)T + (-7.73e9 + 3.72e9i)T^{2} \)
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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

−14.74507996199765010149725224336, −14.05684552020812437079510622082, −12.71428168643335795462364170356, −10.66569686144195168356823805114, −9.919219562467624415221586652977, −9.070472935822366277157165816136, −7.33409132386760722711124769321, −5.59767103320057909833993857745, −3.03165636414540511435686221983, −1.86908919831586750755575762994, 1.93442752207629088257090364454, 3.65540958307300781342941618373, 6.12954802590595627140848596726, 7.72637518559865294509710519454, 9.014847520735339829079802951124, 9.220327740116695574063502819879, 11.61455852507009984427780209527, 13.20488726154029960915147963368, 13.70836041540115892967267385909, 14.88702653163198688787407993976

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