Properties

Degree 2
Conductor 43
Sign $-0.336 - 0.941i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.711 + 1.47i)2-s + (20.0 + 1.50i)3-s + (38.2 + 47.9i)4-s + (−40.7 + 132. i)5-s + (−16.5 + 28.6i)6-s + (72.9 − 42.1i)7-s + (−200. + 45.7i)8-s + (−319. − 48.2i)9-s + (−166. − 154. i)10-s + (−897. + 1.12e3i)11-s + (695. + 1.02e3i)12-s + (2.87e3 − 2.66e3i)13-s + (10.3 + 137. i)14-s + (−1.01e3 + 2.59e3i)15-s + (−798. + 3.49e3i)16-s + (−4.39e3 + 1.35e3i)17-s + ⋯
L(s)  = 1  + (−0.0889 + 0.184i)2-s + (0.743 + 0.0557i)3-s + (0.597 + 0.748i)4-s + (−0.325 + 1.05i)5-s + (−0.0764 + 0.132i)6-s + (0.212 − 0.122i)7-s + (−0.391 + 0.0893i)8-s + (−0.438 − 0.0661i)9-s + (−0.166 − 0.154i)10-s + (−0.674 + 0.845i)11-s + (0.402 + 0.590i)12-s + (1.30 − 1.21i)13-s + (0.00376 + 0.0502i)14-s + (−0.301 + 0.767i)15-s + (−0.194 + 0.853i)16-s + (−0.894 + 0.275i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.336 - 0.941i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.336 - 0.941i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(1.15836 + 1.64491i\)
\(L(\frac12)\)  \(\approx\)  \(1.15836 + 1.64491i\)
\(L(4)\)   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 \( 1 + (-6.51e4 + 4.55e4i)T \)
good2 \( 1 + (0.711 - 1.47i)T + (-39.9 - 50.0i)T^{2} \)
3 \( 1 + (-20.0 - 1.50i)T + (720. + 108. i)T^{2} \)
5 \( 1 + (40.7 - 132. i)T + (-1.29e4 - 8.80e3i)T^{2} \)
7 \( 1 + (-72.9 + 42.1i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (897. - 1.12e3i)T + (-3.94e5 - 1.72e6i)T^{2} \)
13 \( 1 + (-2.87e3 + 2.66e3i)T + (3.60e5 - 4.81e6i)T^{2} \)
17 \( 1 + (4.39e3 - 1.35e3i)T + (1.99e7 - 1.35e7i)T^{2} \)
19 \( 1 + (-1.46e3 - 9.73e3i)T + (-4.49e7 + 1.38e7i)T^{2} \)
23 \( 1 + (-1.54e3 - 3.93e3i)T + (-1.08e8 + 1.00e8i)T^{2} \)
29 \( 1 + (-4.18e4 + 3.13e3i)T + (5.88e8 - 8.86e7i)T^{2} \)
31 \( 1 + (-3.84e4 + 2.62e4i)T + (3.24e8 - 8.26e8i)T^{2} \)
37 \( 1 + (4.51e4 + 2.60e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (-8.81e4 - 4.24e4i)T + (2.96e9 + 3.71e9i)T^{2} \)
47 \( 1 + (8.43e4 + 1.05e5i)T + (-2.39e9 + 1.05e10i)T^{2} \)
53 \( 1 + (-3.70e4 - 3.44e4i)T + (1.65e9 + 2.21e10i)T^{2} \)
59 \( 1 + (-1.68e4 + 7.38e4i)T + (-3.80e10 - 1.83e10i)T^{2} \)
61 \( 1 + (-1.25e5 + 1.84e5i)T + (-1.88e10 - 4.79e10i)T^{2} \)
67 \( 1 + (-3.49e4 + 5.26e3i)T + (8.64e10 - 2.66e10i)T^{2} \)
71 \( 1 + (-3.12e5 - 1.22e5i)T + (9.39e10 + 8.71e10i)T^{2} \)
73 \( 1 + (2.19e5 + 2.36e5i)T + (-1.13e10 + 1.50e11i)T^{2} \)
79 \( 1 + (-3.46e5 - 6.00e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-1.23e4 + 1.64e5i)T + (-3.23e11 - 4.87e10i)T^{2} \)
89 \( 1 + (1.64e5 + 1.23e4i)T + (4.91e11 + 7.40e10i)T^{2} \)
97 \( 1 + (-5.51e5 + 6.91e5i)T + (-1.85e11 - 8.12e11i)T^{2} \)
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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

−15.26939835571930475127832630367, −14.05880306259033760028586928654, −12.75388568185062464855265741352, −11.35970372650755012025723842365, −10.35612450950777960675309422916, −8.378133545826159214487909782595, −7.70488288335900505128028415174, −6.24830420244785909107429800621, −3.61701064782658833710619376704, −2.54768297628614550060868235009, 0.914956939658001680780740714456, 2.68169523108070296947584126214, 4.84024229756012465560842385512, 6.48461006139524812611145249565, 8.472427178281014422910554377998, 9.013945412560049865609900114965, 10.88681952464348963084900690193, 11.73749875261327568165731625848, 13.41813700812448372040172049473, 14.18497944494726463052411176109

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