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Properties

Label 1-1375-1375.1021-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.751 - 0.659i$
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  + (−0.992 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.751 - 0.659i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.751 - 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2240124630 - 0.08441983661i\)
\(L(\frac12)\) \(\approx\) \(0.2240124630 - 0.08441983661i\)
\(L(1)\) \(\approx\) \(0.4549766865 + 0.1626565078i\)
\(L(1)\) \(\approx\) \(0.4549766865 + 0.1626565078i\)

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

−20.38512741592464010354530291948, −19.90525699897759178551556963022, −19.37321335142537450990352601255, −18.785935825024259276666023389708, −17.75563591614616286822948993096, −17.34397469529038113366867914341, −16.70227592468435323277611766496, −15.70752352316064639357920848526, −14.99547824575430325217116497306, −13.98255123620122135811398954651, −13.21148211571267167018871944771, −12.38734553443413614140325870486, −11.6993207540237650944497648234, −10.743789142887162535405374919825, −10.01501919542656511959704725336, −9.148096031029731360420259266212, −8.33981295096300266191570718338, −7.579753708279958617655829112594, −6.72852119325751352750255506258, −6.444729925711509732997959497098, −5.273332231347283027106747044261, −3.82910215485698789202494972301, −2.61019194077978970828770334947, −2.079628107977641758075869014536, −0.76444605106404509628266940857, 0.16678066800657843553129025851, 2.070207259105577312639318075699, 2.70952402184804394984324557152, 3.66825098697177932568176552931, 4.67910815288474663972961154485, 5.87140193489242736428899935198, 6.46612276375644795609936857367, 7.61958864187027354583486984207, 8.477677901792633969494349490953, 9.28781470723261075255344256585, 9.72281282938116463898329629073, 10.46771251650033596329996345532, 11.29218661436399675087686756861, 12.09596453071202582723992315577, 12.80828801800935607395509300671, 14.13025777489315537099715684124, 15.06566249324064606087963223836, 15.492963548911560764460337540396, 16.54330978612340867743211695906, 16.62515906120617106128607559825, 17.754927897782067773108530961297, 18.47611621869725403315848841598, 19.36899860158307152455823221122, 19.97587544971686562884226362009, 20.50165986773863302933478981682

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