# Properties

 Degree $2$ Conductor $63$ Sign $0.128 + 0.991i$ Motivic weight $3$ Primitive yes Self-dual no Analytic rank $0$

# Learn more about

## Dirichlet series

 L(s)  = 1 + (0.648 + 0.374i)2-s + (−3.71 − 6.44i)4-s + (5.42 − 9.40i)5-s + (−18.2 − 3.24i)7-s − 11.5i·8-s + (7.04 − 4.06i)10-s + (44.9 − 25.9i)11-s − 32.1i·13-s + (−10.6 − 8.93i)14-s + (−25.4 + 44.0i)16-s + (40.7 + 70.5i)17-s + (−0.0420 − 0.0242i)19-s − 80.7·20-s + 38.8·22-s + (−77.3 − 44.6i)23-s + ⋯
 L(s)  = 1 + (0.229 + 0.132i)2-s + (−0.464 − 0.805i)4-s + (0.485 − 0.840i)5-s + (−0.984 − 0.175i)7-s − 0.511i·8-s + (0.222 − 0.128i)10-s + (1.23 − 0.710i)11-s − 0.686i·13-s + (−0.202 − 0.170i)14-s + (−0.397 + 0.688i)16-s + (0.581 + 1.00i)17-s + (−0.000507 − 0.000293i)19-s − 0.902·20-s + 0.376·22-s + (−0.701 − 0.404i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $0.128 + 0.991i$ Motivic weight: $$3$$ Character: $\chi_{63} (26, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 63,\ (\ :3/2),\ 0.128 + 0.991i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.04128 - 0.914989i$$ $$L(\frac12)$$ $$\approx$$ $$1.04128 - 0.914989i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1 + (18.2 + 3.24i)T$$
good2 $$1 + (-0.648 - 0.374i)T + (4 + 6.92i)T^{2}$$
5 $$1 + (-5.42 + 9.40i)T + (-62.5 - 108. i)T^{2}$$
11 $$1 + (-44.9 + 25.9i)T + (665.5 - 1.15e3i)T^{2}$$
13 $$1 + 32.1iT - 2.19e3T^{2}$$
17 $$1 + (-40.7 - 70.5i)T + (-2.45e3 + 4.25e3i)T^{2}$$
19 $$1 + (0.0420 + 0.0242i)T + (3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (77.3 + 44.6i)T + (6.08e3 + 1.05e4i)T^{2}$$
29 $$1 - 175. iT - 2.43e4T^{2}$$
31 $$1 + (-186. + 107. i)T + (1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (32.2 - 55.8i)T + (-2.53e4 - 4.38e4i)T^{2}$$
41 $$1 - 411.T + 6.89e4T^{2}$$
43 $$1 + 234.T + 7.95e4T^{2}$$
47 $$1 + (-316. + 547. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + (-230. + 132. i)T + (7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (-175. - 304. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (673. + 389. i)T + (1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-98.0 - 169. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 142. iT - 3.57e5T^{2}$$
73 $$1 + (-676. + 390. i)T + (1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (644. - 1.11e3i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + 235.T + 5.71e5T^{2}$$
89 $$1 + (335. - 580. i)T + (-3.52e5 - 6.10e5i)T^{2}$$
97 $$1 - 655. iT - 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

−14.08706675763107700937279972811, −13.22357722173491868200651529078, −12.30528379524961251408221943667, −10.51159914766809116738846062375, −9.556617431308785597935754923213, −8.597211828284721350519078589856, −6.44441220496694974222684194660, −5.54836456492403106069464060564, −3.87433447861295001991420833508, −0.987657281397556252075140643053, 2.75381209992742197203586368941, 4.19985560806196076935180483389, 6.24860906627905229445656377195, 7.36026656623996728625383306394, 9.147717841234617301778253762807, 9.909231739688911523603659506398, 11.66370136192165741391498839023, 12.41376264739665538200845533916, 13.77689706811468323108089315113, 14.32202058985250479820696334923