Properties

Degree 2
Conductor 43
Sign $0.541 - 0.840i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.93 − 12.3i)2-s + (−47.2 − 3.54i)3-s + (−76.8 − 96.3i)4-s + (8.18 − 26.5i)5-s + (−324. + 561. i)6-s + (−116. + 67.0i)7-s + (−789. + 180. i)8-s + (1.49e3 + 225. i)9-s + (−278. − 258. i)10-s + (1.02e3 − 1.29e3i)11-s + (3.28e3 + 4.82e3i)12-s + (−1.88e3 + 1.74e3i)13-s + (137. + 1.83e3i)14-s + (−480. + 1.22e3i)15-s + (−711. + 3.11e3i)16-s + (−5.98e3 + 1.84e3i)17-s + ⋯
L(s)  = 1  + (0.742 − 1.54i)2-s + (−1.74 − 0.131i)3-s + (−1.20 − 1.50i)4-s + (0.0654 − 0.212i)5-s + (−1.50 + 2.59i)6-s + (−0.338 + 0.195i)7-s + (−1.54 + 0.352i)8-s + (2.05 + 0.309i)9-s + (−0.278 − 0.258i)10-s + (0.773 − 0.969i)11-s + (1.90 + 2.79i)12-s + (−0.856 + 0.794i)13-s + (0.0499 + 0.666i)14-s + (−0.142 + 0.362i)15-s + (−0.173 + 0.760i)16-s + (−1.21 + 0.375i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.541 - 0.840i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.541 - 0.840i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.113794 + 0.0620813i\)
\(L(\frac12)\)  \(\approx\)  \(0.113794 + 0.0620813i\)
\(L(4)\)   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 \( 1 + (-7.58e4 - 2.39e4i)T \)
good2 \( 1 + (-5.93 + 12.3i)T + (-39.9 - 50.0i)T^{2} \)
3 \( 1 + (47.2 + 3.54i)T + (720. + 108. i)T^{2} \)
5 \( 1 + (-8.18 + 26.5i)T + (-1.29e4 - 8.80e3i)T^{2} \)
7 \( 1 + (116. - 67.0i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (-1.02e3 + 1.29e3i)T + (-3.94e5 - 1.72e6i)T^{2} \)
13 \( 1 + (1.88e3 - 1.74e3i)T + (3.60e5 - 4.81e6i)T^{2} \)
17 \( 1 + (5.98e3 - 1.84e3i)T + (1.99e7 - 1.35e7i)T^{2} \)
19 \( 1 + (-1.45e3 - 9.68e3i)T + (-4.49e7 + 1.38e7i)T^{2} \)
23 \( 1 + (3.68e3 + 9.37e3i)T + (-1.08e8 + 1.00e8i)T^{2} \)
29 \( 1 + (4.03e4 - 3.02e3i)T + (5.88e8 - 8.86e7i)T^{2} \)
31 \( 1 + (2.92e3 - 1.99e3i)T + (3.24e8 - 8.26e8i)T^{2} \)
37 \( 1 + (-5.21e3 - 3.01e3i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (8.25e4 + 3.97e4i)T + (2.96e9 + 3.71e9i)T^{2} \)
47 \( 1 + (3.93e4 + 4.93e4i)T + (-2.39e9 + 1.05e10i)T^{2} \)
53 \( 1 + (1.00e5 + 9.31e4i)T + (1.65e9 + 2.21e10i)T^{2} \)
59 \( 1 + (3.70e3 - 1.62e4i)T + (-3.80e10 - 1.83e10i)T^{2} \)
61 \( 1 + (-1.29e5 + 1.90e5i)T + (-1.88e10 - 4.79e10i)T^{2} \)
67 \( 1 + (1.65e5 - 2.49e4i)T + (8.64e10 - 2.66e10i)T^{2} \)
71 \( 1 + (-6.47e5 - 2.54e5i)T + (9.39e10 + 8.71e10i)T^{2} \)
73 \( 1 + (2.43e5 + 2.62e5i)T + (-1.13e10 + 1.50e11i)T^{2} \)
79 \( 1 + (4.70e4 + 8.14e4i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-3.02e4 + 4.03e5i)T + (-3.23e11 - 4.87e10i)T^{2} \)
89 \( 1 + (2.86e5 + 2.15e4i)T + (4.91e11 + 7.40e10i)T^{2} \)
97 \( 1 + (7.30e5 - 9.16e5i)T + (-1.85e11 - 8.12e11i)T^{2} \)
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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

−13.05618753442977085512022338049, −12.26534011965307394311478659656, −11.46589192594170479613817613578, −10.68211848805446226253385731146, −9.407600336515452162895560644471, −6.49284948300670326765118490492, −5.29486461411252368703473613959, −3.99843277334450259961720276560, −1.62945337827140889959257780957, −0.06041371314491650539356363290, 4.40834652842663253052116482493, 5.33328675060356622677957808824, 6.63904823339566307222658069753, 7.21889218734181189432733206170, 9.564002998693597372545851837327, 11.13876752534570663139063412209, 12.42426593277886445954041029656, 13.32793024628058868579595394908, 14.96342666381556288348517061233, 15.66479668981314862374113330661

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