Properties

 Degree $2$ Conductor $4$ Sign $-0.892 + 0.451i$ Motivic weight $32$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−1.52e4 − 6.37e4i)2-s + 6.99e7i·3-s + (−3.83e9 + 1.94e9i)4-s − 1.46e11·5-s + (4.45e12 − 1.06e12i)6-s + 5.71e13i·7-s + (1.82e14 + 2.14e14i)8-s − 3.03e15·9-s + (2.23e15 + 9.34e15i)10-s + 6.39e16i·11-s + (−1.35e17 − 2.67e17i)12-s + 3.82e17·13-s + (3.64e18 − 8.70e17i)14-s − 1.02e19i·15-s + (1.09e19 − 1.48e19i)16-s − 4.07e19·17-s + ⋯
 L(s)  = 1 + (−0.232 − 0.972i)2-s + 1.62i·3-s + (−0.892 + 0.451i)4-s − 0.961·5-s + (1.57 − 0.377i)6-s + 1.72i·7-s + (0.646 + 0.762i)8-s − 1.63·9-s + (0.223 + 0.934i)10-s + 1.39i·11-s + (−0.733 − 1.44i)12-s + 0.574·13-s + (1.67 − 0.399i)14-s − 1.56i·15-s + (0.591 − 0.806i)16-s − 0.837·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(33-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$4$$    =    $$2^{2}$$ Sign: $-0.892 + 0.451i$ Motivic weight: $$32$$ Character: $\chi_{4} (3, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 4,\ (\ :16),\ -0.892 + 0.451i)$$

Particular Values

 $$L(\frac{33}{2})$$ $$\approx$$ $$0.6833074930$$ $$L(\frac12)$$ $$\approx$$ $$0.6833074930$$ $$L(17)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.52e4 + 6.37e4i)T$$
good3 $$1 - 6.99e7iT - 1.85e15T^{2}$$
5 $$1 + 1.46e11T + 2.32e22T^{2}$$
7 $$1 - 5.71e13iT - 1.10e27T^{2}$$
11 $$1 - 6.39e16iT - 2.11e33T^{2}$$
13 $$1 - 3.82e17T + 4.42e35T^{2}$$
17 $$1 + 4.07e19T + 2.36e39T^{2}$$
19 $$1 + 6.13e19iT - 8.31e40T^{2}$$
23 $$1 + 2.81e20iT - 3.76e43T^{2}$$
29 $$1 - 9.24e22T + 6.26e46T^{2}$$
31 $$1 - 4.23e23iT - 5.29e47T^{2}$$
37 $$1 + 4.96e23T + 1.52e50T^{2}$$
41 $$1 - 7.78e25T + 4.06e51T^{2}$$
43 $$1 - 5.67e24iT - 1.86e52T^{2}$$
47 $$1 + 2.70e26iT - 3.21e53T^{2}$$
53 $$1 - 1.23e27T + 1.50e55T^{2}$$
59 $$1 + 3.43e27iT - 4.64e56T^{2}$$
61 $$1 - 4.92e28T + 1.35e57T^{2}$$
67 $$1 - 1.86e29iT - 2.71e58T^{2}$$
71 $$1 + 7.39e29iT - 1.73e59T^{2}$$
73 $$1 - 1.01e30T + 4.22e59T^{2}$$
79 $$1 - 2.63e30iT - 5.29e60T^{2}$$
83 $$1 - 5.56e30iT - 2.57e61T^{2}$$
89 $$1 - 1.36e31T + 2.40e62T^{2}$$
97 $$1 + 9.37e31T + 3.77e63T^{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

−17.93694737542076059508087521050, −15.86697467472792754229819686330, −15.00527860695863652128409944625, −12.28251731241257821486258525797, −11.13904515185557166172191475630, −9.586966661611920501660562857574, −8.556198466636276862970903620399, −5.02035783725721725390619618172, −3.91476788377801770756115578943, −2.43750937467631057657562525146, 0.31902615374232653937731624382, 0.991190937236505938020415310219, 3.89282474232847998717877210938, 6.30695703575476155786899823395, 7.42226052487948717552396843433, 8.290827934663313274010594840215, 11.09613909222827471163569703726, 13.20828309805895705368932460508, 14.01466868335186720086198847632, 16.20011391145279698403496147570