Properties

Degree 2
Conductor 43
Sign $0.431 + 0.902i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.62 + 5.45i)2-s + (−13.6 − 1.02i)3-s + (17.0 + 21.4i)4-s + (29.1 − 94.5i)5-s + (41.3 − 71.6i)6-s + (−270. + 156. i)7-s + (−539. + 123. i)8-s + (−535. − 80.7i)9-s + (438. + 407. i)10-s + (1.16e3 − 1.46e3i)11-s + (−210. − 309. i)12-s + (2.31e3 − 2.14e3i)13-s + (−141. − 1.88e3i)14-s + (−494. + 1.25e3i)15-s + (354. − 1.55e3i)16-s + (308. − 95.2i)17-s + ⋯
L(s)  = 1  + (−0.328 + 0.681i)2-s + (−0.505 − 0.0378i)3-s + (0.266 + 0.334i)4-s + (0.233 − 0.756i)5-s + (0.191 − 0.331i)6-s + (−0.789 + 0.455i)7-s + (−1.05 + 0.240i)8-s + (−0.735 − 0.110i)9-s + (0.438 + 0.407i)10-s + (0.874 − 1.09i)11-s + (−0.122 − 0.179i)12-s + (1.05 − 0.978i)13-s + (−0.0515 − 0.687i)14-s + (−0.146 + 0.373i)15-s + (0.0865 − 0.379i)16-s + (0.0628 − 0.0193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.431 + 0.902i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.431 + 0.902i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.623061 - 0.392768i\)
\(L(\frac12)\)  \(\approx\)  \(0.623061 - 0.392768i\)
\(L(4)\)   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 \( 1 + (6.01e4 - 5.19e4i)T \)
good2 \( 1 + (2.62 - 5.45i)T + (-39.9 - 50.0i)T^{2} \)
3 \( 1 + (13.6 + 1.02i)T + (720. + 108. i)T^{2} \)
5 \( 1 + (-29.1 + 94.5i)T + (-1.29e4 - 8.80e3i)T^{2} \)
7 \( 1 + (270. - 156. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (-1.16e3 + 1.46e3i)T + (-3.94e5 - 1.72e6i)T^{2} \)
13 \( 1 + (-2.31e3 + 2.14e3i)T + (3.60e5 - 4.81e6i)T^{2} \)
17 \( 1 + (-308. + 95.2i)T + (1.99e7 - 1.35e7i)T^{2} \)
19 \( 1 + (472. + 3.13e3i)T + (-4.49e7 + 1.38e7i)T^{2} \)
23 \( 1 + (8.27e3 + 2.10e4i)T + (-1.08e8 + 1.00e8i)T^{2} \)
29 \( 1 + (3.20e4 - 2.40e3i)T + (5.88e8 - 8.86e7i)T^{2} \)
31 \( 1 + (-9.17e3 + 6.25e3i)T + (3.24e8 - 8.26e8i)T^{2} \)
37 \( 1 + (5.14e4 + 2.96e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (-6.26e4 - 3.01e4i)T + (2.96e9 + 3.71e9i)T^{2} \)
47 \( 1 + (7.82e4 + 9.81e4i)T + (-2.39e9 + 1.05e10i)T^{2} \)
53 \( 1 + (-6.10e4 - 5.66e4i)T + (1.65e9 + 2.21e10i)T^{2} \)
59 \( 1 + (-6.36e4 + 2.78e5i)T + (-3.80e10 - 1.83e10i)T^{2} \)
61 \( 1 + (1.23e5 - 1.81e5i)T + (-1.88e10 - 4.79e10i)T^{2} \)
67 \( 1 + (5.84e5 - 8.81e4i)T + (8.64e10 - 2.66e10i)T^{2} \)
71 \( 1 + (-4.91e5 - 1.92e5i)T + (9.39e10 + 8.71e10i)T^{2} \)
73 \( 1 + (1.53e5 + 1.65e5i)T + (-1.13e10 + 1.50e11i)T^{2} \)
79 \( 1 + (3.27e5 + 5.67e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (1.38e4 - 1.84e5i)T + (-3.23e11 - 4.87e10i)T^{2} \)
89 \( 1 + (-1.03e6 - 7.73e4i)T + (4.91e11 + 7.40e10i)T^{2} \)
97 \( 1 + (-2.37e5 + 2.98e5i)T + (-1.85e11 - 8.12e11i)T^{2} \)
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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

−14.75228874478808721030211976313, −13.15659379812050729353682285170, −12.14616852268410603960528113441, −11.02217014393963769466385451895, −9.024818565645114701713959978806, −8.403423592389862041148660094639, −6.39921793561428520612199536786, −5.74196911495942522509829958717, −3.21104153333677076556620597981, −0.40219567369908096407807228867, 1.69097140227707049782462189852, 3.54228341808656194714471004349, 5.99886117025582799701152093797, 6.88132497007361416650133704606, 9.234233234868569620428160979634, 10.19974396530018730356131596228, 11.24986979200090413055685064864, 12.06639933337775427491086084079, 13.77002774655438516056966906417, 14.84228085501397826628083643034

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