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Properties

Label 1-1375-1375.279-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.114 - 0.993i$
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  + (0.425 − 0.904i)2-s + (0.929 − 0.368i)3-s + (−0.637 − 0.770i)4-s + (0.0627 − 0.998i)6-s + (0.809 + 0.587i)7-s + (−0.968 + 0.248i)8-s + (0.728 − 0.684i)9-s + (−0.876 − 0.481i)12-s + (−0.728 + 0.684i)13-s + (0.876 − 0.481i)14-s + (−0.187 + 0.982i)16-s + (0.929 + 0.368i)17-s + (−0.309 − 0.951i)18-s + (0.0627 − 0.998i)19-s + (0.968 + 0.248i)21-s + ⋯
L(s)  = 1  + (0.425 − 0.904i)2-s + (0.929 − 0.368i)3-s + (−0.637 − 0.770i)4-s + (0.0627 − 0.998i)6-s + (0.809 + 0.587i)7-s + (−0.968 + 0.248i)8-s + (0.728 − 0.684i)9-s + (−0.876 − 0.481i)12-s + (−0.728 + 0.684i)13-s + (0.876 − 0.481i)14-s + (−0.187 + 0.982i)16-s + (0.929 + 0.368i)17-s + (−0.309 − 0.951i)18-s + (0.0627 − 0.998i)19-s + (0.968 + 0.248i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.114 - 0.993i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.114 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.904444536 - 2.135919560i\)
\(L(\frac12)\) \(\approx\) \(1.904444536 - 2.135919560i\)
\(L(1)\) \(\approx\) \(1.512662158 - 1.021259041i\)
\(L(1)\) \(\approx\) \(1.512662158 - 1.021259041i\)

Euler product

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

−21.02977370189439321699379579276, −20.556610548000980978869636497188, −19.63087491350146852826749279574, −18.668749061096942567053366344089, −17.9217734855143020516498491991, −17.05135473603064926415875010142, −16.38170631501371498100285048770, −15.6372174017792888967811618501, −14.73349805762178786594868136715, −14.29263387928223497714270220230, −13.87238082518579367887883723784, −12.68914532537057746983524491908, −12.23164437392179292127620822579, −10.79370742257802956558238707416, −10.056580546346291493373558934642, −9.18601133314301935466426402341, −8.21005788297126410818635397784, −7.782656729649477731392290880, −7.12436947456839532885327911634, −5.908587397982954455803452916208, −4.87560269376614079841576006435, −4.43374103578767277937804136025, −3.36943023974135787424560744335, −2.659439852339833954387618567052, −1.181358449179267412856777763, 1.02554783168160207160318778877, 1.981534388841576003712069699832, 2.57922603364437786116766098023, 3.52297922978448367171578656837, 4.475719464002406338968735823644, 5.22127738270145240214627255814, 6.30146602761496457675712789257, 7.40704334293381583285449277454, 8.26960528215864317613075163608, 9.076767755201080061304302931863, 9.67477943644531857278681182026, 10.58308934870012876786921376716, 11.80945842982954241826496824221, 11.96895381327762992289263409712, 12.99030887636832760380864187823, 13.84329827151993931465000161540, 14.25615335041830426149991037065, 15.1549385912570436445658154591, 15.5607623307619497414105970390, 17.23169963970424779593861151111, 17.786231188609758893073005166186, 18.81028134834093034470105404921, 19.16855535285868826284449383587, 19.83326727118118575689024418839, 20.77419868400658078699869411108

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