Properties

Label 1-1375-1375.413-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: $\chi_{1375} (413, \cdot )$
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

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

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