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Properties

Label 1-51-51.20-r0-0-0
Degree $1$
Conductor $51$
Sign $0.978 - 0.204i$
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 + (0.382 − 0.923i)5-s + (−0.382 − 0.923i)7-s + (0.707 + 0.707i)8-s + (0.382 + 0.923i)10-s + (0.923 − 0.382i)11-s + i·13-s + (0.923 + 0.382i)14-s − 16-s + (0.707 − 0.707i)19-s + (−0.923 − 0.382i)20-s + (−0.382 + 0.923i)22-s + (−0.923 + 0.382i)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 + (0.382 − 0.923i)5-s + (−0.382 − 0.923i)7-s + (0.707 + 0.707i)8-s + (0.382 + 0.923i)10-s + (0.923 − 0.382i)11-s + i·13-s + (0.923 + 0.382i)14-s − 16-s + (0.707 − 0.707i)19-s + (−0.923 − 0.382i)20-s + (−0.382 + 0.923i)22-s + (−0.923 + 0.382i)23-s + (−0.707 − 0.707i)25-s + (−0.707 − 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.978 - 0.204i$
Analytic conductor: \(0.236843\)
Root analytic conductor: \(0.236843\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 51,\ (0:\ ),\ 0.978 - 0.204i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6422971005 - 0.06654144815i\)
\(L(\frac12)\) \(\approx\) \(0.6422971005 - 0.06654144815i\)
\(L(1)\) \(\approx\) \(0.7620575145 + 0.01144992888i\)
\(L(1)\) \(\approx\) \(0.7620575145 + 0.01144992888i\)

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 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + iT \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (-0.923 + 0.382i)T \)
29 \( 1 + (-0.382 + 0.923i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
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 \( 1 + (0.382 + 0.923i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.923 - 0.382i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.382 - 0.923i)T \)
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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

−33.963624863128703659136886425838, −32.39172656955133629325101437768, −30.93453089127906769123467870657, −30.10421497787977387555148132097, −29.071493709927190311922076545012, −27.93824736635855733236106774782, −26.9130042109762610823806052741, −25.64257100177467439007286200849, −24.96607134774232818625710116350, −22.50138686741207894383508428738, −22.13941808991746811235623839395, −20.64982521756100334518083453398, −19.37976867488770354327272414669, −18.3606542891573468562317933240, −17.493948646885837867029172205410, −15.893574542803742974517553646565, −14.42547262407925089824357953043, −12.74400064264778801391656763734, −11.61413315627227856141281206454, −10.22696643579947789845017066664, −9.24633477547262585698717682051, −7.62706380484569886950337321570, −6.046134243508905603972434008340, −3.52700503790398479843443663314, −2.15923565641919628858196631835, 1.31456830312164006279266676540, 4.3569764422372164374959043140, 6.01360098702963944515865622564, 7.32839385904376053694182451755, 8.92804305474790553488902391856, 9.76666529811018956690511820504, 11.43426748560959967037499820450, 13.373348510216713533974361906382, 14.3669441944656154508416238826, 16.243219349831302474533948028955, 16.7264206884954923211367367412, 17.92852180599872612659552351042, 19.51843518402446275254752822329, 20.2704722403072661337798529060, 22.004811289707893837447533704001, 23.64364147337621759277428220146, 24.33744482490195987490997937964, 25.57871947188682455971753964017, 26.57774100549286290877116336537, 27.74461773940516819775928774381, 28.80110632129319448879880221828, 29.77220635140881924926484955616, 31.69570702502696844662355249953, 32.81329705525815498685765841543, 33.30936238106885603192412137554

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