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Properties

Label 1-1375-1375.1159-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.310 - 0.950i$
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  + (0.929 − 0.368i)2-s + (0.187 − 0.982i)3-s + (0.728 − 0.684i)4-s + (−0.187 − 0.982i)6-s + (−0.309 + 0.951i)7-s + (0.425 − 0.904i)8-s + (−0.929 − 0.368i)9-s + (−0.535 − 0.844i)12-s + (0.929 + 0.368i)13-s + (0.0627 + 0.998i)14-s + (0.0627 − 0.998i)16-s + (0.425 − 0.904i)17-s − 18-s + (0.876 − 0.481i)19-s + (0.876 + 0.481i)21-s + ⋯
L(s)  = 1  + (0.929 − 0.368i)2-s + (0.187 − 0.982i)3-s + (0.728 − 0.684i)4-s + (−0.187 − 0.982i)6-s + (−0.309 + 0.951i)7-s + (0.425 − 0.904i)8-s + (−0.929 − 0.368i)9-s + (−0.535 − 0.844i)12-s + (0.929 + 0.368i)13-s + (0.0627 + 0.998i)14-s + (0.0627 − 0.998i)16-s + (0.425 − 0.904i)17-s − 18-s + (0.876 − 0.481i)19-s + (0.876 + 0.481i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.310 - 0.950i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.310 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.813341631 - 2.499419916i\)
\(L(\frac12)\) \(\approx\) \(1.813341631 - 2.499419916i\)
\(L(1)\) \(\approx\) \(1.658275368 - 1.083186312i\)
\(L(1)\) \(\approx\) \(1.658275368 - 1.083186312i\)

Euler product

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

−21.1521144386762763991019024989, −20.4728168180203372460266216866, −19.98795477968766537090636920170, −19.03632961887121480891076192715, −17.68701470822604451384840433339, −16.883038609720864895022480718731, −16.46697445731824929510724678350, −15.57076533548954841528051145853, −15.11892480624446688480384561090, −14.102783073757196536780320559, −13.67485709082822718864576791322, −12.85531193578496130613381291772, −11.846308858126770989358036271241, −10.922297823913242867463674480478, −10.46200461762754116409959188793, −9.48320841077789984137015843887, −8.37405943023517299346377796243, −7.7346106437397022527993150462, −6.70396086626274428472959399375, −5.79173702142121965551954170204, −5.12137915396893291805378274808, −4.029278067317425649808116441379, −3.61973882883986836488960153527, −2.85151726951491338953242312198, −1.35555956933744261083196668578, 0.91126188084643901399703717463, 1.84671373984404989273076491240, 2.87781763231747441321008660384, 3.2653046757051239272472548005, 4.69202993933162516717337107477, 5.582844566305123791183691470318, 6.239011542282709952695696447890, 7.00060816830440624851053368000, 7.86235005359943526339398307931, 9.05106708549549268535804265483, 9.5713184076942704710175372889, 11.10732948143052947663309131771, 11.460614697210270128306888926446, 12.29534696035970735627850397083, 13.00823382206769619062875854627, 13.50570039634947031938644823448, 14.43589220013321940608252897218, 14.95953004465558446457704438903, 16.02991067530126077890469187815, 16.52903179000239371700441840060, 18.0895669072367831636607853521, 18.38694247294754816338005829370, 19.19428651633511989060050355034, 19.85852047577800240071526207350, 20.69028232017166628571097635055

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