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Properties

Label 1-275-275.103-r1-0-0
Degree $1$
Conductor $275$
Sign $0.831 + 0.555i$
Analytic cond. $29.5528$
Root an. cond. $29.5528$
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  + (−0.951 − 0.309i)2-s + (0.951 + 0.309i)3-s + (0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s + (0.587 + 0.809i)7-s + (−0.587 − 0.809i)8-s + (0.809 + 0.587i)9-s + (0.587 + 0.809i)12-s + (0.587 − 0.809i)13-s + (−0.309 − 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.587 + 0.809i)17-s + (−0.587 − 0.809i)18-s + (0.809 − 0.587i)19-s + (0.309 + 0.951i)21-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.951 + 0.309i)3-s + (0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s + (0.587 + 0.809i)7-s + (−0.587 − 0.809i)8-s + (0.809 + 0.587i)9-s + (0.587 + 0.809i)12-s + (0.587 − 0.809i)13-s + (−0.309 − 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.587 + 0.809i)17-s + (−0.587 − 0.809i)18-s + (0.809 − 0.587i)19-s + (0.309 + 0.951i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.831 + 0.555i$
Analytic conductor: \(29.5528\)
Root analytic conductor: \(29.5528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (1:\ ),\ 0.831 + 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.994424907 + 0.6052330257i\)
\(L(\frac12)\) \(\approx\) \(1.994424907 + 0.6052330257i\)
\(L(1)\) \(\approx\) \(1.166901108 + 0.1325201229i\)
\(L(1)\) \(\approx\) \(1.166901108 + 0.1325201229i\)

Euler product

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

−25.41631285928136248961891137522, −24.659166804685987556120123237781, −23.93430920370659281359762613965, −23.019679125144880888880478015421, −21.13942090182017696806647459967, −20.68131102875200369570590021654, −19.75961333640471678218016192405, −18.96828669400692437500258604920, −18.08404705286569764274104574798, −17.27783903650700538259705107299, −16.08468641494065791329076908308, −15.335834694276124446198570798655, −14.05436130069821121396192584367, −13.74194650048302318288895860262, −12.005056369796059264628963168655, −11.039437093947038496148139611566, −9.879627246304724593307102657593, −9.05611565539321682983214609837, −8.071119191235750884633134313774, −7.311496599898567235979289023491, −6.42337591221708222391092201414, −4.71349284159276325299367093628, −3.27076715844723324754583853161, −1.88087629017774953807434494090, −0.92570603190762682292386178723, 1.21320650565365310247395970961, 2.440118490818198994590852353944, 3.29675292032315376778599213140, 4.790515141671022462438736839716, 6.388720150266371679652495547732, 7.72731219925770876759692798844, 8.51161123640348646852294083074, 9.10498977391607353720780497397, 10.30469199534948779938134022499, 11.08642994587153088722166423494, 12.33077646564721679319732526109, 13.29272021115066837473293584428, 14.66727956597200182736539587882, 15.47589863616323246130862896070, 16.14925624846708614383108458595, 17.612404930955045234293701559220, 18.21956440385836826623302752701, 19.258392549375567382961380219627, 19.95393935890587110579689517015, 20.929977019783657244232635299262, 21.4397846438774662192580940225, 22.52776754394264206060018084931, 24.38121228157808755157327616216, 24.74226759963804672946050700767, 25.760901314381065696355344346621

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