Properties

Degree 2
Conductor 43
Sign $0.987 + 0.158i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.10 − 10.6i)2-s + (−7.80 + 16.2i)3-s + (−46.4 + 58.2i)4-s + (−223. − 50.9i)5-s + 211.·6-s − 193. i·7-s + (121. + 27.6i)8-s + (252. + 316. i)9-s + (600. + 2.62e3i)10-s + (1.25e3 + 1.56e3i)11-s + (−581. − 1.20e3i)12-s + (771. − 3.38e3i)13-s + (−2.04e3 + 986. i)14-s + (2.56e3 − 3.22e3i)15-s + (736. + 3.22e3i)16-s + (314. + 1.37e3i)17-s + ⋯
L(s)  = 1  + (−0.638 − 1.32i)2-s + (−0.289 + 0.600i)3-s + (−0.726 + 0.910i)4-s + (−1.78 − 0.407i)5-s + 0.980·6-s − 0.563i·7-s + (0.236 + 0.0539i)8-s + (0.346 + 0.434i)9-s + (0.600 + 2.62i)10-s + (0.940 + 1.17i)11-s + (−0.336 − 0.699i)12-s + (0.351 − 1.53i)13-s + (−0.746 + 0.359i)14-s + (0.761 − 0.954i)15-s + (0.179 + 0.787i)16-s + (0.0640 + 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.987 + 0.158i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.987 + 0.158i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.585330 - 0.0465351i\)
\(L(\frac12)\)  \(\approx\)  \(0.585330 - 0.0465351i\)
\(L(4)\)   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 \( 1 + (-7.41e4 + 2.86e4i)T \)
good2 \( 1 + (5.10 + 10.6i)T + (-39.9 + 50.0i)T^{2} \)
3 \( 1 + (7.80 - 16.2i)T + (-454. - 569. i)T^{2} \)
5 \( 1 + (223. + 50.9i)T + (1.40e4 + 6.77e3i)T^{2} \)
7 \( 1 + 193. iT - 1.17e5T^{2} \)
11 \( 1 + (-1.25e3 - 1.56e3i)T + (-3.94e5 + 1.72e6i)T^{2} \)
13 \( 1 + (-771. + 3.38e3i)T + (-4.34e6 - 2.09e6i)T^{2} \)
17 \( 1 + (-314. - 1.37e3i)T + (-2.17e7 + 1.04e7i)T^{2} \)
19 \( 1 + (-223. - 178. i)T + (1.04e7 + 4.58e7i)T^{2} \)
23 \( 1 + (5.13e3 + 6.44e3i)T + (-3.29e7 + 1.44e8i)T^{2} \)
29 \( 1 + (-1.92e4 - 3.99e4i)T + (-3.70e8 + 4.65e8i)T^{2} \)
31 \( 1 + (1.08e4 - 5.21e3i)T + (5.53e8 - 6.93e8i)T^{2} \)
37 \( 1 - 1.11e4iT - 2.56e9T^{2} \)
41 \( 1 + (6.01e4 - 2.89e4i)T + (2.96e9 - 3.71e9i)T^{2} \)
47 \( 1 + (2.29e4 - 2.87e4i)T + (-2.39e9 - 1.05e10i)T^{2} \)
53 \( 1 + (-5.17e4 - 2.26e5i)T + (-1.99e10 + 9.61e9i)T^{2} \)
59 \( 1 + (-2.23e4 - 9.78e4i)T + (-3.80e10 + 1.83e10i)T^{2} \)
61 \( 1 + (-9.51e4 + 1.97e5i)T + (-3.21e10 - 4.02e10i)T^{2} \)
67 \( 1 + (2.55e5 - 3.20e5i)T + (-2.01e10 - 8.81e10i)T^{2} \)
71 \( 1 + (9.47e4 + 7.55e4i)T + (2.85e10 + 1.24e11i)T^{2} \)
73 \( 1 + (-5.68e4 - 1.29e4i)T + (1.36e11 + 6.56e10i)T^{2} \)
79 \( 1 - 7.49e5T + 2.43e11T^{2} \)
83 \( 1 + (4.04e5 + 1.94e5i)T + (2.03e11 + 2.55e11i)T^{2} \)
89 \( 1 + (-1.64e5 + 3.42e5i)T + (-3.09e11 - 3.88e11i)T^{2} \)
97 \( 1 + (-5.43e4 - 6.81e4i)T + (-1.85e11 + 8.12e11i)T^{2} \)
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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

−14.98839873298298757387931575094, −12.73712647060223342809469047485, −12.07875547247496352315440901611, −10.88320077132116501288889513641, −10.23180257473770997719879409283, −8.691767410840646412246342137929, −7.46070796389250519543189474657, −4.53034279342901756937151505998, −3.54076483208771928018711149075, −1.03025495179294177394132625449, 0.48053023210994825366921487336, 3.87465783598605989469439627507, 6.23134266337560808882220993729, 7.03200074203503267764285656601, 8.161897103350598010381535121287, 9.168642322350154085667130004407, 11.63813941325224318915042013935, 11.85450056498484685466183367171, 14.01416071651798553435915203922, 15.14237591947803890989886570549

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