Properties

 Label 2-112-7.6-c6-0-21 Degree $2$ Conductor $112$ Sign $-0.387 - 0.921i$ Analytic cond. $25.7660$ Root an. cond. $5.07602$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 45.1i·3-s − 45.1i·5-s + (−133 − 316. i)7-s − 1.31e3·9-s − 874·11-s + 2.21e3i·13-s − 2.03e3·15-s − 5.96e3i·17-s − 3.11e3i·19-s + (−1.42e4 + 6.00e3i)21-s − 4.73e3·23-s + 1.35e4·25-s + 2.62e4i·27-s + 1.11e4·29-s + 2.74e4i·31-s + ⋯
 L(s)  = 1 − 1.67i·3-s − 0.361i·5-s + (−0.387 − 0.921i)7-s − 1.79·9-s − 0.656·11-s + 1.00i·13-s − 0.604·15-s − 1.21i·17-s − 0.454i·19-s + (−1.54 + 0.648i)21-s − 0.389·23-s + 0.869·25-s + 1.33i·27-s + 0.457·29-s + 0.921i·31-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$112$$    =    $$2^{4} \cdot 7$$ Sign: $-0.387 - 0.921i$ Analytic conductor: $$25.7660$$ Root analytic conductor: $$5.07602$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{112} (97, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 112,\ (\ :3),\ -0.387 - 0.921i)$$

Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.7185793007$$ $$L(\frac12)$$ $$\approx$$ $$0.7185793007$$ $$L(4)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1 + (133 + 316. i)T$$
good3 $$1 + 45.1iT - 729T^{2}$$
5 $$1 + 45.1iT - 1.56e4T^{2}$$
11 $$1 + 874T + 1.77e6T^{2}$$
13 $$1 - 2.21e3iT - 4.82e6T^{2}$$
17 $$1 + 5.96e3iT - 2.41e7T^{2}$$
19 $$1 + 3.11e3iT - 4.70e7T^{2}$$
23 $$1 + 4.73e3T + 1.48e8T^{2}$$
29 $$1 - 1.11e4T + 5.94e8T^{2}$$
31 $$1 - 2.74e4iT - 8.87e8T^{2}$$
37 $$1 - 3.00e3T + 2.56e9T^{2}$$
41 $$1 - 5.75e4iT - 4.75e9T^{2}$$
43 $$1 + 3.14e4T + 6.32e9T^{2}$$
47 $$1 - 7.24e4iT - 1.07e10T^{2}$$
53 $$1 + 7.64e4T + 2.21e10T^{2}$$
59 $$1 + 1.13e5iT - 4.21e10T^{2}$$
61 $$1 + 2.75e5iT - 5.15e10T^{2}$$
67 $$1 + 4.95e5T + 9.04e10T^{2}$$
71 $$1 - 1.84e5T + 1.28e11T^{2}$$
73 $$1 + 6.09e4iT - 1.51e11T^{2}$$
79 $$1 - 5.34e5T + 2.43e11T^{2}$$
83 $$1 - 7.14e5iT - 3.26e11T^{2}$$
89 $$1 + 6.29e5iT - 4.96e11T^{2}$$
97 $$1 - 8.14e5iT - 8.32e11T^{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

−12.00907090358046090597276009343, −10.94713503559953690667192637801, −9.438799918308330870951007759284, −8.162791081241336266117612346859, −7.16141105188838870396674138524, −6.49377077789719751102827339783, −4.82689185657972481967586583733, −2.84915580985799963762364128930, −1.36880019733931573511789669625, −0.24021167838274501703095140520, 2.68195708644248051985688528437, 3.75856884942224993675502341503, 5.18053809759035456296527067338, 6.06413230432099244801668572873, 8.092747967334230954732713323839, 9.072811271749236617184909214262, 10.24517907146753832095005386178, 10.62226182277916047184674247722, 11.99264056979823282148928434181, 13.13993357787200253256655506065