Properties

Degree 2
Conductor 43
Sign $-0.0188 + 0.999i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.294 − 1.28i)2-s + (−1.69 − 1.57i)3-s + (5.63 − 2.71i)4-s + (1.82 − 0.275i)5-s + (−1.52 + 2.64i)6-s + (−11.2 − 19.5i)7-s + (−11.7 − 14.7i)8-s + (−1.61 − 21.5i)9-s + (−0.891 − 2.27i)10-s + (51.6 + 24.8i)11-s + (−13.8 − 4.26i)12-s + (−17.9 + 45.6i)13-s + (−21.8 + 20.3i)14-s + (−3.53 − 2.40i)15-s + (15.6 − 19.6i)16-s + (39.1 + 5.89i)17-s + ⋯
L(s)  = 1  + (−0.103 − 0.455i)2-s + (−0.326 − 0.303i)3-s + (0.704 − 0.339i)4-s + (0.163 − 0.0246i)5-s + (−0.104 + 0.180i)6-s + (−0.609 − 1.05i)7-s + (−0.519 − 0.650i)8-s + (−0.0598 − 0.799i)9-s + (−0.0282 − 0.0718i)10-s + (1.41 + 0.681i)11-s + (−0.332 − 0.102i)12-s + (−0.382 + 0.974i)13-s + (−0.417 + 0.387i)14-s + (−0.0608 − 0.0414i)15-s + (0.244 − 0.306i)16-s + (0.558 + 0.0841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0188 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0188 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.0188 + 0.999i$
motivic weight  =  \(3\)
character  :  $\chi_{43} (10, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ -0.0188 + 0.999i)\)
\(L(2)\)  \(\approx\)  \(0.899706 - 0.916847i\)
\(L(\frac12)\)  \(\approx\)  \(0.899706 - 0.916847i\)
\(L(\frac{5}{2})\)   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 \( 1 + (280. - 28.1i)T \)
good2 \( 1 + (0.294 + 1.28i)T + (-7.20 + 3.47i)T^{2} \)
3 \( 1 + (1.69 + 1.57i)T + (2.01 + 26.9i)T^{2} \)
5 \( 1 + (-1.82 + 0.275i)T + (119. - 36.8i)T^{2} \)
7 \( 1 + (11.2 + 19.5i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-51.6 - 24.8i)T + (829. + 1.04e3i)T^{2} \)
13 \( 1 + (17.9 - 45.6i)T + (-1.61e3 - 1.49e3i)T^{2} \)
17 \( 1 + (-39.1 - 5.89i)T + (4.69e3 + 1.44e3i)T^{2} \)
19 \( 1 + (9.44 - 126. i)T + (-6.78e3 - 1.02e3i)T^{2} \)
23 \( 1 + (-126. + 86.2i)T + (4.44e3 - 1.13e4i)T^{2} \)
29 \( 1 + (6.72 - 6.24i)T + (1.82e3 - 2.43e4i)T^{2} \)
31 \( 1 + (-265. - 81.9i)T + (2.46e4 + 1.67e4i)T^{2} \)
37 \( 1 + (11.4 - 19.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (58.5 + 256. i)T + (-6.20e4 + 2.99e4i)T^{2} \)
47 \( 1 + (-61.6 + 29.6i)T + (6.47e4 - 8.11e4i)T^{2} \)
53 \( 1 + (-113. - 287. i)T + (-1.09e5 + 1.01e5i)T^{2} \)
59 \( 1 + (120. - 150. i)T + (-4.57e4 - 2.00e5i)T^{2} \)
61 \( 1 + (209. - 64.5i)T + (1.87e5 - 1.27e5i)T^{2} \)
67 \( 1 + (-18.7 + 250. i)T + (-2.97e5 - 4.48e4i)T^{2} \)
71 \( 1 + (-776. - 529. i)T + (1.30e5 + 3.33e5i)T^{2} \)
73 \( 1 + (-124. + 317. i)T + (-2.85e5 - 2.64e5i)T^{2} \)
79 \( 1 + (435. + 753. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-710. - 659. i)T + (4.27e4 + 5.70e5i)T^{2} \)
89 \( 1 + (320. + 297. i)T + (5.26e4 + 7.02e5i)T^{2} \)
97 \( 1 + (889. + 428. i)T + (5.69e5 + 7.13e5i)T^{2} \)
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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

−15.05973277219763574572925531574, −14.07413155656211159106065014497, −12.36994408426750401123169613368, −11.80382506167635518549031396283, −10.28594007244600841225611833304, −9.430386003245188151144521417622, −7.02723317565691166007450911352, −6.35467582435011459452506565537, −3.78477270096843951778322385276, −1.30917056674018379710708499670, 2.87267320881136228013815832595, 5.45163192284618199129335181743, 6.59217625411148019208324359189, 8.200428542564316048753850415889, 9.538192113488399908868293095598, 11.18314810338121008536605066341, 12.00075741473878629795065909618, 13.44730563581332546037762287694, 15.06298029775336761959537550477, 15.74967774766883872321254859941

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