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Properties

Label 1-1375-1375.1102-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.950 - 0.312i$
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  + (0.844 − 0.535i)2-s + (−0.982 + 0.187i)3-s + (0.425 − 0.904i)4-s + (−0.728 + 0.684i)6-s + (−0.951 − 0.309i)7-s + (−0.125 − 0.992i)8-s + (0.929 − 0.368i)9-s + (−0.248 + 0.968i)12-s + (−0.368 − 0.929i)13-s + (−0.968 + 0.248i)14-s + (−0.637 − 0.770i)16-s + (0.982 + 0.187i)17-s + (0.587 − 0.809i)18-s + (0.728 − 0.684i)19-s + (0.992 + 0.125i)21-s + ⋯
L(s)  = 1  + (0.844 − 0.535i)2-s + (−0.982 + 0.187i)3-s + (0.425 − 0.904i)4-s + (−0.728 + 0.684i)6-s + (−0.951 − 0.309i)7-s + (−0.125 − 0.992i)8-s + (0.929 − 0.368i)9-s + (−0.248 + 0.968i)12-s + (−0.368 − 0.929i)13-s + (−0.968 + 0.248i)14-s + (−0.637 − 0.770i)16-s + (0.982 + 0.187i)17-s + (0.587 − 0.809i)18-s + (0.728 − 0.684i)19-s + (0.992 + 0.125i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.950 - 0.312i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.950 - 0.312i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1986157683 - 1.241365332i\)
\(L(\frac12)\) \(\approx\) \(0.1986157683 - 1.241365332i\)
\(L(1)\) \(\approx\) \(0.9231138494 - 0.5948567576i\)
\(L(1)\) \(\approx\) \(0.9231138494 - 0.5948567576i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.844 - 0.535i)T \)
3 \( 1 + (-0.982 + 0.187i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (-0.368 - 0.929i)T \)
17 \( 1 + (0.982 + 0.187i)T \)
19 \( 1 + (0.728 - 0.684i)T \)
23 \( 1 + (0.770 + 0.637i)T \)
29 \( 1 + (0.876 - 0.481i)T \)
31 \( 1 + (-0.992 + 0.125i)T \)
37 \( 1 + (-0.368 - 0.929i)T \)
41 \( 1 + (0.929 - 0.368i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (-0.684 + 0.728i)T \)
53 \( 1 + (0.684 - 0.728i)T \)
59 \( 1 + (-0.0627 - 0.998i)T \)
61 \( 1 + (-0.0627 + 0.998i)T \)
67 \( 1 + (-0.982 - 0.187i)T \)
71 \( 1 + (-0.187 - 0.982i)T \)
73 \( 1 + (-0.844 + 0.535i)T \)
79 \( 1 + (-0.992 - 0.125i)T \)
83 \( 1 + (-0.125 - 0.992i)T \)
89 \( 1 + (-0.968 + 0.248i)T \)
97 \( 1 + (-0.904 - 0.425i)T \)
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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

−21.55537896544162132703407347314, −20.71505746950011266385629300611, −19.652723370916013221415523610907, −18.714124288125446309053811888634, −18.1654651331164615864215618190, −16.96870357613634750958292617495, −16.536152422030631504277867687485, −16.10271226795823484619433194646, −15.13880332098867656657016138790, −14.29957274762234649592180563663, −13.45489283480199918096019576170, −12.6621943549237116448588446757, −12.09246743370247010101493873959, −11.560704659974840236889044322095, −10.44159211913732056587904397812, −9.62307907006081759203510696052, −8.5468369319826907443573482873, −7.3488616790066154156803267419, −6.88049338432292603806664778928, −6.063284740038160758412115453447, −5.38515515892237118293846651168, −4.60227213837596512079684605828, −3.59630467364864666295349373979, −2.69894324574969280413632447510, −1.39850824965952180647859624279, 0.43576181117638244302406406394, 1.393188346748669096024303251100, 2.90712376947295379411409167289, 3.493362408843067968291365970426, 4.51287633248549200368981057035, 5.41368378442457538071890044992, 5.87393973320777653913798971235, 6.9051758708236424346958200007, 7.49534783103777324363446611750, 9.25194180291256900831038312910, 9.95000285594659191968504208053, 10.5041949696630859268162497697, 11.32609268994044707806370928153, 12.101493268428206982789018175089, 12.81764257757177744723079924862, 13.26091839638594342033918657546, 14.34467245185920398809537840274, 15.23742799409632355044536799757, 15.91949082972549097854149687176, 16.50552750745382812635880582347, 17.49925546953874349968293080293, 18.235319669338022600261210301309, 19.29208668322362453009687695745, 19.68590394105728875947395665094, 20.71234882124710410651691626395

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