Properties

 Degree 2 Conductor 43 Sign $-0.285 + 0.958i$ Motivic weight 4 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 − 4.43i·2-s + 6.93i·3-s − 3.69·4-s − 22.3i·5-s + 30.7·6-s − 51.9i·7-s − 54.6i·8-s + 32.8·9-s − 99.2·10-s − 25.0·11-s − 25.6i·12-s − 38.7·13-s − 230.·14-s + 155.·15-s − 301.·16-s + 111.·17-s + ⋯
 L(s)  = 1 − 1.10i·2-s + 0.771i·3-s − 0.230·4-s − 0.894i·5-s + 0.855·6-s − 1.06i·7-s − 0.853i·8-s + 0.405·9-s − 0.992·10-s − 0.206·11-s − 0.177i·12-s − 0.229·13-s − 1.17·14-s + 0.689·15-s − 1.17·16-s + 0.385·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $-0.285 + 0.958i$ motivic weight = $$4$$ character : $\chi_{43} (42, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :2),\ -0.285 + 0.958i)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$0.944406 - 1.26738i$$ $$L(\frac12)$$ $$\approx$$ $$0.944406 - 1.26738i$$ $$L(3)$$ 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 + (528. - 1.77e3i)T$$
good2 $$1 + 4.43iT - 16T^{2}$$
3 $$1 - 6.93iT - 81T^{2}$$
5 $$1 + 22.3iT - 625T^{2}$$
7 $$1 + 51.9iT - 2.40e3T^{2}$$
11 $$1 + 25.0T + 1.46e4T^{2}$$
13 $$1 + 38.7T + 2.85e4T^{2}$$
17 $$1 - 111.T + 8.35e4T^{2}$$
19 $$1 - 238. iT - 1.30e5T^{2}$$
23 $$1 - 823.T + 2.79e5T^{2}$$
29 $$1 - 424. iT - 7.07e5T^{2}$$
31 $$1 + 1.44e3T + 9.23e5T^{2}$$
37 $$1 - 626. iT - 1.87e6T^{2}$$
41 $$1 - 580.T + 2.82e6T^{2}$$
47 $$1 + 170.T + 4.87e6T^{2}$$
53 $$1 - 4.30e3T + 7.89e6T^{2}$$
59 $$1 + 65.4T + 1.21e7T^{2}$$
61 $$1 + 3.89e3iT - 1.38e7T^{2}$$
67 $$1 + 5.44e3T + 2.01e7T^{2}$$
71 $$1 - 5.84e3iT - 2.54e7T^{2}$$
73 $$1 + 4.77e3iT - 2.83e7T^{2}$$
79 $$1 - 1.01e4T + 3.89e7T^{2}$$
83 $$1 - 4.43e3T + 4.74e7T^{2}$$
89 $$1 + 9.87e3iT - 6.27e7T^{2}$$
97 $$1 + 7.76e3T + 8.85e7T^{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

−14.89151572871071198055180787466, −13.24035799669626406200111739996, −12.49353856907522278478572175468, −11.01282867444372784217424885705, −10.21291827274175245807107861086, −9.174039135997210294741116580549, −7.24263709797992840175899225623, −4.82798709935685033398652664996, −3.58652799255498187144706696335, −1.16354562543860040550095235321, 2.46760618385530248088167326570, 5.44178105415812301238963872529, 6.75679385434071954538918285510, 7.49174550156238721852069130842, 8.967956552969827735958371232942, 10.85777215174713415309861185511, 12.13906988356065896434497465178, 13.42592560371781921677256805500, 14.81301042167504246204230075432, 15.29112426904277435635926363677