Properties

Label 1-1375-1375.1039-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.604 + 0.796i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (0.637 − 0.770i)2-s + (−0.876 + 0.481i)3-s + (−0.187 − 0.982i)4-s + (−0.187 + 0.982i)6-s + (0.809 − 0.587i)7-s + (−0.876 − 0.481i)8-s + (0.535 − 0.844i)9-s + (0.637 + 0.770i)12-s + (−0.535 + 0.844i)13-s + (0.0627 − 0.998i)14-s + (−0.929 + 0.368i)16-s + (0.425 + 0.904i)17-s + (−0.309 − 0.951i)18-s + (−0.992 + 0.125i)19-s + (−0.425 + 0.904i)21-s + ⋯
L(s)  = 1  + (0.637 − 0.770i)2-s + (−0.876 + 0.481i)3-s + (−0.187 − 0.982i)4-s + (−0.187 + 0.982i)6-s + (0.809 − 0.587i)7-s + (−0.876 − 0.481i)8-s + (0.535 − 0.844i)9-s + (0.637 + 0.770i)12-s + (−0.535 + 0.844i)13-s + (0.0627 − 0.998i)14-s + (−0.929 + 0.368i)16-s + (0.425 + 0.904i)17-s + (−0.309 − 0.951i)18-s + (−0.992 + 0.125i)19-s + (−0.425 + 0.904i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.604 + 0.796i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.604 + 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8111995283 + 0.4028432496i\)
\(L(\frac12)\) \(\approx\) \(0.8111995283 + 0.4028432496i\)
\(L(1)\) \(\approx\) \(0.9455981854 - 0.2164229090i\)
\(L(1)\) \(\approx\) \(0.9455981854 - 0.2164229090i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.637 - 0.770i)T \)
3 \( 1 + (-0.876 + 0.481i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (-0.535 + 0.844i)T \)
17 \( 1 + (0.425 + 0.904i)T \)
19 \( 1 + (-0.992 + 0.125i)T \)
23 \( 1 + (-0.0627 + 0.998i)T \)
29 \( 1 + (-0.425 + 0.904i)T \)
31 \( 1 + (-0.992 + 0.125i)T \)
37 \( 1 + (0.637 + 0.770i)T \)
41 \( 1 + (-0.929 + 0.368i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (0.425 - 0.904i)T \)
53 \( 1 + (0.187 + 0.982i)T \)
59 \( 1 + (-0.929 + 0.368i)T \)
61 \( 1 + (-0.637 + 0.770i)T \)
67 \( 1 + (0.992 - 0.125i)T \)
71 \( 1 + (-0.187 - 0.982i)T \)
73 \( 1 + (-0.535 - 0.844i)T \)
79 \( 1 + (0.728 + 0.684i)T \)
83 \( 1 + (-0.728 + 0.684i)T \)
89 \( 1 + (-0.637 + 0.770i)T \)
97 \( 1 + (-0.876 + 0.481i)T \)
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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

−21.00548915037990427686936874806, −20.15695506168454215740015608394, −18.82165640996065793781115494178, −18.28963221520911692117622683490, −17.53685128175173946348816129871, −16.98643692337412689997180532787, −16.21110694663182305582556751322, −15.37309792701810058165225657548, −14.67276124544002700086823145237, −13.9496790121500309815000755740, −12.771053474299890269609741356128, −12.60837098286275970204584448623, −11.583731805235010672143308717266, −11.05972935854130386132040080838, −9.870692121709970627213922709903, −8.69183435715613611850079026879, −7.876631991463802493318410013547, −7.31788660012158847658735371037, −6.315473941583318248859058946223, −5.624351214210664029832375007172, −4.959760036211034417792837035749, −4.26805756865126372789613872686, −2.81979574944707633406521778100, −1.978246317443465790737707124319, −0.315385742793528335000834218962, 1.28225390154641981175232363466, 1.938294071137015964862858728, 3.50186140684430836058986308555, 4.11558666886774814043493476679, 4.875584698241917802655498274729, 5.57152567386437112977396442326, 6.49749458947638233163782575078, 7.363143343869862202144514437947, 8.71863809427983987485230055265, 9.60317255798349966030256255583, 10.4786774703308718670196796201, 10.8743925870384077408235225573, 11.73473273474021245412408078422, 12.269801977589025891694225489955, 13.18147985529413140872840492658, 14.05412047301743100065198082311, 14.92269397696073472597019721645, 15.27188179048928127185652968276, 16.7549483272499342655411142807, 16.91762811550827028515366211864, 18.05268198384596234318419592927, 18.664345991371780390543199038579, 19.6603658332537236128856585383, 20.339565414725885123088260881261, 21.24453634317909111328860302940

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