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Properties

Label 1-1375-1375.1146-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.892 + 0.451i$
Analytic cond. $147.764$
Root an. cond. $147.764$
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  + (0.992 + 0.125i)2-s + (0.0627 + 0.998i)3-s + (0.968 + 0.248i)4-s + (−0.0627 + 0.998i)6-s + (0.809 − 0.587i)7-s + (0.929 + 0.368i)8-s + (−0.992 + 0.125i)9-s + (−0.187 + 0.982i)12-s + (0.992 − 0.125i)13-s + (0.876 − 0.481i)14-s + (0.876 + 0.481i)16-s + (0.929 + 0.368i)17-s − 18-s + (0.637 − 0.770i)19-s + (0.637 + 0.770i)21-s + ⋯
L(s)  = 1  + (0.992 + 0.125i)2-s + (0.0627 + 0.998i)3-s + (0.968 + 0.248i)4-s + (−0.0627 + 0.998i)6-s + (0.809 − 0.587i)7-s + (0.929 + 0.368i)8-s + (−0.992 + 0.125i)9-s + (−0.187 + 0.982i)12-s + (0.992 − 0.125i)13-s + (0.876 − 0.481i)14-s + (0.876 + 0.481i)16-s + (0.929 + 0.368i)17-s − 18-s + (0.637 − 0.770i)19-s + (0.637 + 0.770i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.892 + 0.451i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.892 + 0.451i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.824610139 + 1.388708963i\)
\(L(\frac12)\) \(\approx\) \(5.824610139 + 1.388708963i\)
\(L(1)\) \(\approx\) \(2.383822826 + 0.6304020374i\)
\(L(1)\) \(\approx\) \(2.383822826 + 0.6304020374i\)

Euler product

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

−20.53688553848121075715409811161, −20.116595473519598248445645273642, −19.03044184412541144625029809203, −18.4099353948900495848951163405, −17.824520770085198827618691814641, −16.61222099255184080209935126235, −16.00100768082320334134185930749, −14.97368736212771266059725716146, −14.11646692915376138514045439168, −13.97740967712528844205433868549, −12.81731448752684699820493663225, −12.2409355818978628102822740924, −11.585542133269684658507492597461, −10.97899801385206378985275313876, −9.831484047798398196805200313452, −8.561991792094853449793248933986, −7.88409736768173073160329152014, −7.13968265451062311712507182198, −6.034333405504371909557443630064, −5.67388804697662582493294726590, −4.67949496244421582986936433297, −3.47918800576467931360719418293, −2.75771640020003903732969958197, −1.60656049668033841076575311716, −1.19021268874288538445030741515, 0.83219621729205176491406766783, 2.124778391947479711541602212977, 3.1525785604707713018136991754, 4.11006162370330328857605353056, 4.38551164846033954380064024679, 5.67026075760880684899565482858, 5.91980622623746505324644572909, 7.42772819350210771095309691542, 7.96539176237598012669905301622, 8.99441400478181903353098979586, 10.13233209797043270423616879864, 10.783360019638924178899311611124, 11.42165893759446776406602560575, 12.136926412982123588645071700075, 13.34112017368180537754634868717, 14.00091740023891218879323164276, 14.47442452069856266971651709940, 15.49931985993265207691282757517, 15.79912596029384873176313783674, 16.85509531271853353597388529399, 17.26275004075076458145872748310, 18.39669403633021932709758245051, 19.61496886484372084392993076260, 20.35551745439808177636151543955, 20.81374853335876434802063029517

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