Properties

Degree $2$
Conductor $43$
Sign $0.863 + 0.504i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.75 − 5.96i)2-s + (−3.77 + 4.73i)3-s + (−5.82 + 25.5i)4-s + (−49.8 + 24.0i)5-s + 46.1·6-s + 187.·7-s + (−39.9 + 19.2i)8-s + (45.9 + 201. i)9-s + (380. + 183. i)10-s + (−98.0 − 429. i)11-s + (−98.8 − 123. i)12-s + (551. − 265. i)13-s + (−893. − 1.12e3i)14-s + (74.5 − 326. i)15-s + (1.05e3 + 510. i)16-s + (1.02e3 + 494. i)17-s + ⋯
L(s)  = 1  + (−0.840 − 1.05i)2-s + (−0.242 + 0.303i)3-s + (−0.182 + 0.797i)4-s + (−0.892 + 0.429i)5-s + 0.523·6-s + 1.44·7-s + (−0.220 + 0.106i)8-s + (0.188 + 0.827i)9-s + (1.20 + 0.579i)10-s + (−0.244 − 1.06i)11-s + (−0.198 − 0.248i)12-s + (0.904 − 0.435i)13-s + (−1.21 − 1.52i)14-s + (0.0855 − 0.374i)15-s + (1.03 + 0.498i)16-s + (0.862 + 0.415i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.863 + 0.504i$
Motivic weight: \(5\)
Character: $\chi_{43} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :5/2),\ 0.863 + 0.504i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.906843 - 0.245796i\)
\(L(\frac12)\) \(\approx\) \(0.906843 - 0.245796i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (1.12e4 + 4.47e3i)T \)
good2 \( 1 + (4.75 + 5.96i)T + (-7.12 + 31.1i)T^{2} \)
3 \( 1 + (3.77 - 4.73i)T + (-54.0 - 236. i)T^{2} \)
5 \( 1 + (49.8 - 24.0i)T + (1.94e3 - 2.44e3i)T^{2} \)
7 \( 1 - 187.T + 1.68e4T^{2} \)
11 \( 1 + (98.0 + 429. i)T + (-1.45e5 + 6.98e4i)T^{2} \)
13 \( 1 + (-551. + 265. i)T + (2.31e5 - 2.90e5i)T^{2} \)
17 \( 1 + (-1.02e3 - 494. i)T + (8.85e5 + 1.11e6i)T^{2} \)
19 \( 1 + (-459. + 2.01e3i)T + (-2.23e6 - 1.07e6i)T^{2} \)
23 \( 1 + (-846. - 3.70e3i)T + (-5.79e6 + 2.79e6i)T^{2} \)
29 \( 1 + (-5.09e3 - 6.39e3i)T + (-4.56e6 + 1.99e7i)T^{2} \)
31 \( 1 + (-3.39e3 - 4.25e3i)T + (-6.37e6 + 2.79e7i)T^{2} \)
37 \( 1 - 1.10e4T + 6.93e7T^{2} \)
41 \( 1 + (974. + 1.22e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
47 \( 1 + (-3.17e3 + 1.39e4i)T + (-2.06e8 - 9.95e7i)T^{2} \)
53 \( 1 + (1.49e4 + 7.22e3i)T + (2.60e8 + 3.26e8i)T^{2} \)
59 \( 1 + (-2.15e4 - 1.03e4i)T + (4.45e8 + 5.58e8i)T^{2} \)
61 \( 1 + (1.58e4 - 1.99e4i)T + (-1.87e8 - 8.23e8i)T^{2} \)
67 \( 1 + (-1.35e3 + 5.92e3i)T + (-1.21e9 - 5.85e8i)T^{2} \)
71 \( 1 + (-5.77e3 + 2.53e4i)T + (-1.62e9 - 7.82e8i)T^{2} \)
73 \( 1 + (2.40e4 - 1.15e4i)T + (1.29e9 - 1.62e9i)T^{2} \)
79 \( 1 - 2.81e4T + 3.07e9T^{2} \)
83 \( 1 + (-1.95e3 + 2.45e3i)T + (-8.76e8 - 3.84e9i)T^{2} \)
89 \( 1 + (-6.73e4 + 8.44e4i)T + (-1.24e9 - 5.44e9i)T^{2} \)
97 \( 1 + (-2.22e4 - 9.72e4i)T + (-7.73e9 + 3.72e9i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.99135364801753604307035544427, −13.56490355537889151297168488230, −11.68891859376865023060226141605, −11.13640285926817104453047973764, −10.47432609415440071151963094835, −8.616263461970048424113135784529, −7.79385561873312847907162032895, −5.25389282334520696889486741612, −3.26499806495442475028250308422, −1.18168202112961928379006355595, 0.936480812269946355072238807735, 4.41128120904797371287161364072, 6.27084883399250574515265430369, 7.74963414152716841203908591921, 8.259951031811348908979195127537, 9.798041399594506412052328796065, 11.65775653908852567934095684826, 12.37443369024262627347587185261, 14.43795415049015493249338965113, 15.26040312318002393368516634751

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