Properties

 Label 2-43-43.6-c5-0-13 Degree $2$ Conductor $43$ Sign $0.698 + 0.715i$ Analytic cond. $6.89650$ Root an. cond. $2.62611$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + 7.09·2-s + (1.12 − 1.94i)3-s + 18.4·4-s + (51.2 − 88.8i)5-s + (7.96 − 13.8i)6-s + (−36.3 − 62.9i)7-s − 96.5·8-s + (118. + 206. i)9-s + (364. − 630. i)10-s + 469.·11-s + (20.6 − 35.7i)12-s + (235. + 408. i)13-s + (−257. − 446. i)14-s + (−115. − 199. i)15-s − 1.27e3·16-s + (−406. − 704. i)17-s + ⋯
 L(s)  = 1 + 1.25·2-s + (0.0720 − 0.124i)3-s + 0.575·4-s + (0.917 − 1.58i)5-s + (0.0903 − 0.156i)6-s + (−0.280 − 0.485i)7-s − 0.533·8-s + (0.489 + 0.848i)9-s + (1.15 − 1.99i)10-s + 1.16·11-s + (0.0414 − 0.0717i)12-s + (0.386 + 0.669i)13-s + (−0.351 − 0.609i)14-s + (−0.132 − 0.228i)15-s − 1.24·16-s + (−0.341 − 0.591i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$43$$ Sign: $0.698 + 0.715i$ Analytic conductor: $$6.89650$$ Root analytic conductor: $$2.62611$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{43} (6, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 43,\ (\ :5/2),\ 0.698 + 0.715i)$$

Particular Values

 $$L(3)$$ $$\approx$$ $$2.99186 - 1.26112i$$ $$L(\frac12)$$ $$\approx$$ $$2.99186 - 1.26112i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad43 $$1 + (1.21e4 + 673. i)T$$
good2 $$1 - 7.09T + 32T^{2}$$
3 $$1 + (-1.12 + 1.94i)T + (-121.5 - 210. i)T^{2}$$
5 $$1 + (-51.2 + 88.8i)T + (-1.56e3 - 2.70e3i)T^{2}$$
7 $$1 + (36.3 + 62.9i)T + (-8.40e3 + 1.45e4i)T^{2}$$
11 $$1 - 469.T + 1.61e5T^{2}$$
13 $$1 + (-235. - 408. i)T + (-1.85e5 + 3.21e5i)T^{2}$$
17 $$1 + (406. + 704. i)T + (-7.09e5 + 1.22e6i)T^{2}$$
19 $$1 + (548. - 950. i)T + (-1.23e6 - 2.14e6i)T^{2}$$
23 $$1 + (1.42e3 - 2.47e3i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 + (-3.90e3 - 6.76e3i)T + (-1.02e7 + 1.77e7i)T^{2}$$
31 $$1 + (-1.20e3 + 2.09e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (3.72e3 - 6.45e3i)T + (-3.46e7 - 6.00e7i)T^{2}$$
41 $$1 - 1.52e4T + 1.15e8T^{2}$$
47 $$1 + 1.53e4T + 2.29e8T^{2}$$
53 $$1 + (-4.74e3 + 8.22e3i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 - 3.48e4T + 7.14e8T^{2}$$
61 $$1 + (2.14e4 + 3.72e4i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (1.94e4 - 3.37e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + (3.44e3 + 5.96e3i)T + (-9.02e8 + 1.56e9i)T^{2}$$
73 $$1 + (-1.01e4 - 1.76e4i)T + (-1.03e9 + 1.79e9i)T^{2}$$
79 $$1 + (1.09e4 + 1.89e4i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 + (-6.52e3 + 1.13e4i)T + (-1.96e9 - 3.41e9i)T^{2}$$
89 $$1 + (1.67e4 - 2.90e4i)T + (-2.79e9 - 4.83e9i)T^{2}$$
97 $$1 - 6.64e4T + 8.58e9T^{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

−14.20556015725717604365275983570, −13.58050954593030181308900034951, −12.83199312685329916145089563684, −11.77437938192274179222137093600, −9.771436916096250668110022265097, −8.687030835585357123539356095239, −6.53357107609108379278774165108, −5.14862332172017506344408972528, −4.11871714647600580729781473164, −1.56874379605996310487298028437, 2.65311686918505634773454570688, 3.95026690438349065196504713762, 6.13078248592484609638062358917, 6.53909051945534269717579000600, 9.148431459965204429730796362142, 10.38029975938669406044299180265, 11.79160692692501709356068845773, 12.97766446489658571204166112376, 14.08218909708358866805470005824, 14.83387520601216526723139896812