# Properties

 Label 1-51-51.5-r0-0-0 Degree $1$ Conductor $51$ Sign $0.581 - 0.813i$ Analytic cond. $0.236843$ Root an. cond. $0.236843$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.707 − 0.707i)2-s − i·4-s + (0.923 + 0.382i)5-s + (−0.923 + 0.382i)7-s + (−0.707 − 0.707i)8-s + (0.923 − 0.382i)10-s + (−0.382 − 0.923i)11-s + i·13-s + (−0.382 + 0.923i)14-s − 16-s + (−0.707 + 0.707i)19-s + (0.382 − 0.923i)20-s + (−0.923 − 0.382i)22-s + (0.382 + 0.923i)23-s + (0.707 + 0.707i)25-s + (0.707 + 0.707i)26-s + ⋯
 L(s)  = 1 + (0.707 − 0.707i)2-s − i·4-s + (0.923 + 0.382i)5-s + (−0.923 + 0.382i)7-s + (−0.707 − 0.707i)8-s + (0.923 − 0.382i)10-s + (−0.382 − 0.923i)11-s + i·13-s + (−0.382 + 0.923i)14-s − 16-s + (−0.707 + 0.707i)19-s + (0.382 − 0.923i)20-s + (−0.923 − 0.382i)22-s + (0.382 + 0.923i)23-s + (0.707 + 0.707i)25-s + (0.707 + 0.707i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$51$$    =    $$3 \cdot 17$$ Sign: $0.581 - 0.813i$ Analytic conductor: $$0.236843$$ Root analytic conductor: $$0.236843$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{51} (5, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 51,\ (0:\ ),\ 0.581 - 0.813i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.073468872 - 0.5520118545i$$ $$L(\frac12)$$ $$\approx$$ $$1.073468872 - 0.5520118545i$$ $$L(1)$$ $$\approx$$ $$1.266613724 - 0.4802442762i$$ $$L(1)$$ $$\approx$$ $$1.266613724 - 0.4802442762i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
17 $$1$$
good2 $$1 + (0.707 - 0.707i)T$$
5 $$1 + (0.923 + 0.382i)T$$
7 $$1 + (-0.923 + 0.382i)T$$
11 $$1 + (-0.382 - 0.923i)T$$
13 $$1 + iT$$
19 $$1 + (-0.707 + 0.707i)T$$
23 $$1 + (0.382 + 0.923i)T$$
29 $$1 + (-0.923 - 0.382i)T$$
31 $$1 + (0.382 - 0.923i)T$$
37 $$1 + (-0.382 + 0.923i)T$$
41 $$1 + (0.923 - 0.382i)T$$
43 $$1 + (-0.707 - 0.707i)T$$
47 $$1 - iT$$
53 $$1 + (0.707 - 0.707i)T$$
59 $$1 + (-0.707 - 0.707i)T$$
61 $$1 + (0.923 - 0.382i)T$$
67 $$1 - T$$
71 $$1 + (0.382 - 0.923i)T$$
73 $$1 + (-0.923 - 0.382i)T$$
79 $$1 + (0.382 + 0.923i)T$$
83 $$1 + (-0.707 + 0.707i)T$$
89 $$1 + iT$$
97 $$1 + (0.923 + 0.382i)T$$
show less
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−33.44657892459718591823956084913, −32.67010967736033519444944004969, −31.87730538715759754821937497163, −30.38007337805068288491812346311, −29.457739728642323649962347602648, −28.16866604392292444510531815547, −26.360191340723545165049537328717, −25.562178375112663099648181589258, −24.671247395336254634413870614651, −23.23334101587885699381626431225, −22.386372942915666269657055926558, −21.12207932734039890338739951559, −20.04235366829820696792744399404, −17.97652387124561088080246240653, −17.03064523400416974508473332171, −15.86593016246207879013460799635, −14.58364597001410579844136483665, −13.10257417903394571919068381514, −12.68827574425098473389254005308, −10.40110807701196291794857783270, −8.945172224963068101123202757269, −7.22462671948539322076681634780, −5.98325479894939274924531763629, −4.637312196105888845422004935, −2.76258862654243603655535260519, 2.135722386191857940427111850324, 3.54377474995825108066705513046, 5.56154372592305736792916753549, 6.52158674962710634182971062110, 9.1576494781898835305903582605, 10.219204829410693483338177889311, 11.55004183262477693727561863686, 13.05435965946917157747091379314, 13.8455163029787596907441629515, 15.204669091687296902896811802912, 16.7174484809467345030269993549, 18.6002421835320791686772939227, 19.1984832487482112104598357280, 20.945173842690863390516904235977, 21.70988547910216204481340259455, 22.678640480847322585597992613175, 23.98555342316311807314023184517, 25.27273713032784694495504020420, 26.47956088789980705213727735317, 28.143001296360242391900535016, 29.26686881941384193319214591187, 29.66262670477572750220681194307, 31.28110450094999227856658925336, 32.07984590428003414648483789809, 33.2275228919229621341663299965