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Properties

Label 1-51-51.41-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 $0$

Origins

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Normalization:  

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: $\chi_{51} (41, \cdot )$
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 \( 1 + (0.923 - 0.382i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

Graph of the $Z$-function along the critical line