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Properties

Label 1-1375-1375.4-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  + (0.187 + 0.982i)2-s + (−0.0627 − 0.998i)3-s + (−0.929 + 0.368i)4-s + (0.968 − 0.248i)6-s + (0.809 + 0.587i)7-s + (−0.535 − 0.844i)8-s + (−0.992 + 0.125i)9-s + (0.425 + 0.904i)12-s + (0.992 − 0.125i)13-s + (−0.425 + 0.904i)14-s + (0.728 − 0.684i)16-s + (−0.0627 + 0.998i)17-s + (−0.309 − 0.951i)18-s + (0.968 − 0.248i)19-s + (0.535 − 0.844i)21-s + ⋯
L(s)  = 1  + (0.187 + 0.982i)2-s + (−0.0627 − 0.998i)3-s + (−0.929 + 0.368i)4-s + (0.968 − 0.248i)6-s + (0.809 + 0.587i)7-s + (−0.535 − 0.844i)8-s + (−0.992 + 0.125i)9-s + (0.425 + 0.904i)12-s + (0.992 − 0.125i)13-s + (−0.425 + 0.904i)14-s + (0.728 − 0.684i)16-s + (−0.0627 + 0.998i)17-s + (−0.309 − 0.951i)18-s + (0.968 − 0.248i)19-s + (0.535 − 0.844i)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: $\chi_{1375} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.676 + 0.736i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.491304340 + 0.6552911157i\)
\(L(\frac12)\) \(\approx\) \(1.491304340 + 0.6552911157i\)
\(L(1)\) \(\approx\) \(1.110805578 + 0.3192475001i\)
\(L(1)\) \(\approx\) \(1.110805578 + 0.3192475001i\)

Euler product

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

−20.746722220101562525635217420447, −20.30358653999427973788057739503, −19.563816981742719738158092125628, −18.31031489140381874613961373001, −17.98247644003186830195234748206, −16.99019249948552030432585317600, −16.194639932542961207524129080430, −15.34755933392283570632741976456, −14.38903066324763303644870796378, −13.92347875815710027849924239692, −13.20148035571773004272227718974, −11.875634224486083221256315191082, −11.358295060722007711568655959653, −10.880125735369684167226339583668, −9.86984840218422469611200595357, −9.42636434018949340397830881142, −8.43087803237152893592029023979, −7.66893947888103793184010704095, −6.13272539871140035047513390741, −5.25092992722206575326657958224, −4.596121185019943775174343111985, −3.76245905446749917826477914363, −3.12622916718684551460281115368, −1.886414435797478697512879992705, −0.81502914924778477612226046425, 0.93432884816150647005384165070, 1.99337006467165583860224793232, 3.194281723843085472535125002931, 4.26775870650095037411223616670, 5.360448753213033418775038265954, 5.93619828160865275622108631689, 6.66120323626047219192206877602, 7.62050186548087222043167579923, 8.35404786021868021467934028190, 8.66599524532339430456601866747, 9.90437390216712274027375690579, 11.16375379735997566545377398982, 11.86557518462893075026129607930, 12.714168773148576005843370716893, 13.36509457945840295330919923997, 14.154227544413839665369533224556, 14.72841515596252632593555912344, 15.61348533162311402563608500881, 16.35915443199727755280802111604, 17.3931378247750833761709858433, 17.79418469240367619164295429991, 18.509343421708780062257328933198, 19.03243043747938701870426275056, 20.18094902278467578520452752870, 21.01060559600916047818415301394

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