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Properties

Label 1-1375-1375.1033-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.882 - 0.470i$
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  + (−0.481 + 0.876i)2-s + (0.368 − 0.929i)3-s + (−0.535 − 0.844i)4-s + (0.637 + 0.770i)6-s + (0.951 + 0.309i)7-s + (0.998 − 0.0627i)8-s + (−0.728 − 0.684i)9-s + (−0.982 + 0.187i)12-s + (0.684 − 0.728i)13-s + (−0.728 + 0.684i)14-s + (−0.425 + 0.904i)16-s + (0.844 + 0.535i)17-s + (0.951 − 0.309i)18-s + (−0.929 + 0.368i)19-s + (0.637 − 0.770i)21-s + ⋯
L(s)  = 1  + (−0.481 + 0.876i)2-s + (0.368 − 0.929i)3-s + (−0.535 − 0.844i)4-s + (0.637 + 0.770i)6-s + (0.951 + 0.309i)7-s + (0.998 − 0.0627i)8-s + (−0.728 − 0.684i)9-s + (−0.982 + 0.187i)12-s + (0.684 − 0.728i)13-s + (−0.728 + 0.684i)14-s + (−0.425 + 0.904i)16-s + (0.844 + 0.535i)17-s + (0.951 − 0.309i)18-s + (−0.929 + 0.368i)19-s + (0.637 − 0.770i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.882 - 0.470i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.882 - 0.470i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.460130423 - 0.3651343224i\)
\(L(\frac12)\) \(\approx\) \(1.460130423 - 0.3651343224i\)
\(L(1)\) \(\approx\) \(1.052551935 + 0.02286219472i\)
\(L(1)\) \(\approx\) \(1.052551935 + 0.02286219472i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.481 + 0.876i)T \)
3 \( 1 + (0.368 - 0.929i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
13 \( 1 + (0.684 - 0.728i)T \)
17 \( 1 + (0.844 + 0.535i)T \)
19 \( 1 + (-0.929 + 0.368i)T \)
23 \( 1 + (-0.125 - 0.992i)T \)
29 \( 1 + (0.968 - 0.248i)T \)
31 \( 1 + (0.535 - 0.844i)T \)
37 \( 1 + (0.904 + 0.425i)T \)
41 \( 1 + (0.992 + 0.125i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (-0.998 - 0.0627i)T \)
53 \( 1 + (-0.770 - 0.637i)T \)
59 \( 1 + (0.187 + 0.982i)T \)
61 \( 1 + (0.992 - 0.125i)T \)
67 \( 1 + (0.248 - 0.968i)T \)
71 \( 1 + (0.0627 - 0.998i)T \)
73 \( 1 + (0.982 + 0.187i)T \)
79 \( 1 + (-0.929 - 0.368i)T \)
83 \( 1 + (0.368 + 0.929i)T \)
89 \( 1 + (0.187 - 0.982i)T \)
97 \( 1 + (-0.248 - 0.968i)T \)
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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

−20.90943760799641804805230973640, −20.28753239813480601041038369095, −19.48397390705720876651564002372, −18.84583431744513763511206226679, −17.80491499992076432013532696550, −17.28481918475042957960485957331, −16.385160170913620695478879099253, −15.79531787480257870418487508656, −14.58140141430943305831476649197, −14.05776523673160966511206980462, −13.34088129256427466942555373817, −12.165143857217349279112885213609, −11.32460796001085011340891753075, −10.920614776281540404385125831680, −10.03799300403079284215945279058, −9.35602711719264542249598504298, −8.48655129988903196041996498909, −8.03136155771843643230445226890, −6.934959410887219990915860509481, −5.433767247945211049920185457561, −4.58418421018133850467515132438, −3.96921270128033207666300476224, −3.067917387922002257716740557183, −2.09826926064289345070663733251, −1.106427722668225351097061925863, 0.800191588485416166609012687711, 1.625045371989835602691408459116, 2.65470715431908882845568196464, 4.030771911719631204890860951818, 5.05642771575090115551233137844, 6.119483158875747036765162623278, 6.403986438892582018922697336862, 7.772027033083892259751055540130, 8.12299156795488993840467742036, 8.5905204177161421021058283733, 9.72315336257464226832230502035, 10.63800856604126100564859705445, 11.49695200589228786066412487482, 12.5561605619997089726566468445, 13.21631529880014437297338828216, 14.16168717613530227456263923357, 14.72723158622064380744223986783, 15.2349002065001169794003862137, 16.370032313550830105019231062397, 17.156193188412729274141874724255, 17.85477466391803834188128591234, 18.37863947631371385938963425481, 19.029484701150841545031725733352, 19.7713841813231415108909928603, 20.67657153664763869890288297504

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