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Properties

Label 1-275-275.153-r0-0-0
Degree $1$
Conductor $275$
Sign $0.770 - 0.637i$
Analytic cond. $1.27709$
Root an. cond. $1.27709$
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  + (0.587 − 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.309 + 0.951i)6-s + i·7-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (0.587 + 0.809i)12-s + (0.587 + 0.809i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.951 + 0.309i)17-s − i·18-s + (0.309 − 0.951i)19-s + (−0.309 − 0.951i)21-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.951 + 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.309 + 0.951i)6-s + i·7-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (0.587 + 0.809i)12-s + (0.587 + 0.809i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (0.951 + 0.309i)17-s − i·18-s + (0.309 − 0.951i)19-s + (−0.309 − 0.951i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(1.27709\)
Root analytic conductor: \(1.27709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ 0.770 - 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.148409277 - 0.4134527807i\)
\(L(\frac12)\) \(\approx\) \(1.148409277 - 0.4134527807i\)
\(L(1)\) \(\approx\) \(1.041386717 - 0.3200707264i\)
\(L(1)\) \(\approx\) \(1.041386717 - 0.3200707264i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
7 \( 1 + iT \)
13 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.951 - 0.309i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−25.414906403460189950549584729070, −24.85574088336645154384490945881, −23.69377768446475566246212169598, −23.08924133646821608039015481751, −22.70078796214034658054960695544, −21.397641235650180110196089399234, −20.675897890978609566850140699572, −19.21591561818781005499672129585, −17.9938645259655169913834517758, −17.42439655925094284398792983200, −16.44472776548095874330292100877, −15.90687871307018011478080597383, −14.57159841413481318601270289667, −13.59408189273528608182197368087, −12.84556252709930259088689569739, −11.88122680484844995225747849554, −10.83064099600502530843315675807, −9.725621767503116288088351816376, −7.96079033722463647641426065842, −7.42394506730216540778855376375, −6.25428666225254407514105052177, −5.477000210100481069489225155391, −4.377350151430047906651198378650, −3.25915666879125707042796474944, −1.053789919500835836477975205522, 1.140538205640531771963812389541, 2.60791576325726221259873365855, 3.942986358206453647908769982723, 5.00658171331710240863718960018, 5.82685142496476759152096318067, 6.77476838196457211766843506652, 8.77600897969363193770589916201, 9.643506850653920569596141612450, 10.73647362876799248578786017861, 11.549477386715309389396935034655, 12.25202606367280128014146547432, 13.12699665580437026311976656866, 14.41844083344911668073836594252, 15.35164998224199210528042434628, 16.20196044380470928243569234645, 17.43158978312517863852053212946, 18.59212318140821802964378538859, 18.92142099373267876895780319226, 20.45110858134068685246992435515, 21.27275245107130098218929721776, 21.933991253877543820869782533022, 22.629472608505384066431570686216, 23.664704056465029382610692032557, 24.19892685844702184893573121219, 25.50712401579202053485021906614

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