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Properties

Label 1-1375-1375.1169-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.676 - 0.736i$
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  + (−0.876 − 0.481i)2-s + (0.637 − 0.770i)3-s + (0.535 + 0.844i)4-s + (−0.929 + 0.368i)6-s + (0.809 − 0.587i)7-s + (−0.0627 − 0.998i)8-s + (−0.187 − 0.982i)9-s + (0.992 + 0.125i)12-s + (0.187 + 0.982i)13-s + (−0.992 + 0.125i)14-s + (−0.425 + 0.904i)16-s + (0.637 + 0.770i)17-s + (−0.309 + 0.951i)18-s + (−0.929 + 0.368i)19-s + (0.0627 − 0.998i)21-s + ⋯
L(s)  = 1  + (−0.876 − 0.481i)2-s + (0.637 − 0.770i)3-s + (0.535 + 0.844i)4-s + (−0.929 + 0.368i)6-s + (0.809 − 0.587i)7-s + (−0.0627 − 0.998i)8-s + (−0.187 − 0.982i)9-s + (0.992 + 0.125i)12-s + (0.187 + 0.982i)13-s + (−0.992 + 0.125i)14-s + (−0.425 + 0.904i)16-s + (0.637 + 0.770i)17-s + (−0.309 + 0.951i)18-s + (−0.929 + 0.368i)19-s + (0.0627 − 0.998i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.676 - 0.736i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.676 - 0.736i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.355887619 - 0.5957879198i\)
\(L(\frac12)\) \(\approx\) \(1.355887619 - 0.5957879198i\)
\(L(1)\) \(\approx\) \(0.9531667093 - 0.3713826481i\)
\(L(1)\) \(\approx\) \(0.9531667093 - 0.3713826481i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.876 - 0.481i)T \)
3 \( 1 + (0.637 - 0.770i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (0.187 + 0.982i)T \)
17 \( 1 + (0.637 + 0.770i)T \)
19 \( 1 + (-0.929 + 0.368i)T \)
23 \( 1 + (0.425 + 0.904i)T \)
29 \( 1 + (0.968 - 0.248i)T \)
31 \( 1 + (0.0627 + 0.998i)T \)
37 \( 1 + (0.187 + 0.982i)T \)
41 \( 1 + (-0.187 - 0.982i)T \)
43 \( 1 + (0.809 + 0.587i)T \)
47 \( 1 + (0.929 + 0.368i)T \)
53 \( 1 + (0.929 + 0.368i)T \)
59 \( 1 + (0.728 + 0.684i)T \)
61 \( 1 + (0.728 - 0.684i)T \)
67 \( 1 + (0.637 + 0.770i)T \)
71 \( 1 + (-0.637 + 0.770i)T \)
73 \( 1 + (-0.876 - 0.481i)T \)
79 \( 1 + (0.0627 - 0.998i)T \)
83 \( 1 + (-0.0627 - 0.998i)T \)
89 \( 1 + (-0.992 + 0.125i)T \)
97 \( 1 + (-0.535 - 0.844i)T \)
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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

−20.78547738735011721420489505670, −20.22514134535838823454331521800, −19.359974362086377827976781462474, −18.6444630211020730585195440752, −17.921109054452873674897100108929, −17.12834293877224667970657081672, −16.33438368043858883272090627613, −15.61021661678934730548196048434, −14.939351142092338488196788098753, −14.51559649759299210659739911932, −13.59315041844749175485500711732, −12.394369810298451502692284895995, −11.297387347538716604686359258445, −10.699362983421096127527437306781, −9.968906710250453975110109528005, −9.10462122739560884943002242633, −8.43406134676704973326661890581, −7.95032125499612750995007721726, −6.98199900838252605285022732481, −5.73734187582713914017935953605, −5.156123983874426620071337322791, −4.21790066040916905365054166190, −2.76101470231135326417837064463, −2.27314113236712529568081824709, −0.84623673256306688961224689552, 1.09818676129606577854315855796, 1.62177495054224167829748283638, 2.56103958040455679386530839969, 3.637174457182793294961650651054, 4.3484096192378729742446313949, 5.96589071339757304419942998471, 6.93526159556339014473111106059, 7.47839168671548219427417166021, 8.43682155500807873159220528566, 8.72784785685535469848307767998, 9.86922468496031759742511358858, 10.60870872732657819230106015312, 11.53270312898240745018240721666, 12.14270620583979888206810140533, 12.97854869693741431337847538352, 13.824810570972885926309820037006, 14.51626476539029069875497767944, 15.40969750220899520339741649916, 16.47608176919473159423000705838, 17.3526384991814676543913611669, 17.64124436986173632367390911547, 18.73377280179902908694507893050, 19.16755597436742667363904389207, 19.769007366565698641912515013834, 20.744335458400375870007619383075

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