Properties

 Label 2-105-21.5-c3-0-29 Degree $2$ Conductor $105$ Sign $-0.459 + 0.887i$ Analytic cond. $6.19520$ Root an. cond. $2.48901$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (2.43 + 1.40i)2-s + (−2.49 − 4.55i)3-s + (−0.0626 − 0.108i)4-s + (−2.5 + 4.33i)5-s + (0.319 − 14.5i)6-s + (−12.1 − 13.9i)7-s − 22.8i·8-s + (−14.5 + 22.7i)9-s + (−12.1 + 7.01i)10-s + (−24.0 + 13.9i)11-s + (−0.338 + 0.556i)12-s − 85.9i·13-s + (−10.0 − 50.9i)14-s + (25.9 + 0.569i)15-s + (31.4 − 54.5i)16-s + (18.2 + 31.6i)17-s + ⋯
 L(s)  = 1 + (0.859 + 0.496i)2-s + (−0.480 − 0.876i)3-s + (−0.00783 − 0.0135i)4-s + (−0.223 + 0.387i)5-s + (0.0217 − 0.991i)6-s + (−0.657 − 0.753i)7-s − 1.00i·8-s + (−0.537 + 0.843i)9-s + (−0.384 + 0.221i)10-s + (−0.660 + 0.381i)11-s + (−0.00813 + 0.0133i)12-s − 1.83i·13-s + (−0.191 − 0.973i)14-s + (0.447 + 0.00980i)15-s + (0.492 − 0.852i)16-s + (0.260 + 0.451i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.887i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.459 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $-0.459 + 0.887i$ Analytic conductor: $$6.19520$$ Root analytic conductor: $$2.48901$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{105} (26, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 105,\ (\ :3/2),\ -0.459 + 0.887i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$0.634145 - 1.04253i$$ $$L(\frac12)$$ $$\approx$$ $$0.634145 - 1.04253i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (2.49 + 4.55i)T$$
5 $$1 + (2.5 - 4.33i)T$$
7 $$1 + (12.1 + 13.9i)T$$
good2 $$1 + (-2.43 - 1.40i)T + (4 + 6.92i)T^{2}$$
11 $$1 + (24.0 - 13.9i)T + (665.5 - 1.15e3i)T^{2}$$
13 $$1 + 85.9iT - 2.19e3T^{2}$$
17 $$1 + (-18.2 - 31.6i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (55.3 + 31.9i)T + (3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (-100. - 58.2i)T + (6.08e3 + 1.05e4i)T^{2}$$
29 $$1 - 80.6iT - 2.43e4T^{2}$$
31 $$1 + (-268. + 155. i)T + (1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (41.0 - 71.0i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 + 85.8T + 6.89e4T^{2}$$
43 $$1 + 94.2T + 7.95e4T^{2}$$
47 $$1 + (-199. + 345. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (121. - 70.2i)T + (7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (443. + 768. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (-98.2 - 56.7i)T + (1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-130. - 225. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 390. iT - 3.57e5T^{2}$$
73 $$1 + (183. - 105. i)T + (1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (-529. + 916. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 - 1.00e3T + 5.71e5T^{2}$$
89 $$1 + (490. - 850. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 + 1.69e3iT - 9.12e5T^{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

−13.10241613649376993184003987649, −12.42228596831685668122403925184, −10.79356809502569831218595873156, −10.07870564415266359607356523926, −7.980093518508733577303301750448, −7.03156683928883792843291283022, −6.06867296786011080345334296085, −4.94101952830881961367612794597, −3.18214307236009092166027942561, −0.54274738426863843686560425590, 2.81329539794593940251750383040, 4.18335302342029017566168857003, 5.11955206261205035578099227457, 6.38731432592007607465302571131, 8.508604642855989362368532475178, 9.344656311138076380837881774914, 10.74419812716321317030651277774, 11.81918805608140695192363554071, 12.30924629058009195805309844393, 13.51278463680938350139348591273