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Properties

Label 1-275-275.81-r0-0-0
Degree $1$
Conductor $275$
Sign $0.920 + 0.390i$
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  + (−0.809 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)4-s + 6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)12-s + (0.309 − 0.951i)13-s + 14-s + (−0.809 + 0.587i)16-s + 17-s + (−0.809 + 0.587i)18-s + (0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)4-s + 6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)12-s + (0.309 − 0.951i)13-s + 14-s + (−0.809 + 0.587i)16-s + 17-s + (−0.809 + 0.587i)18-s + (0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.920 + 0.390i$
Analytic conductor: \(1.27709\)
Root analytic conductor: \(1.27709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ 0.920 + 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5358035249 + 0.1088261455i\)
\(L(\frac12)\) \(\approx\) \(0.5358035249 + 0.1088261455i\)
\(L(1)\) \(\approx\) \(0.5572156589 + 0.01158921178i\)
\(L(1)\) \(\approx\) \(0.5572156589 + 0.01158921178i\)

Euler product

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

−25.66492522289357348241506021912, −24.73615159599267806041147778923, −23.825252838399760814898339706, −23.17695697568710537304115456080, −22.410124469989206401391132517882, −20.94487226185938439681154492400, −19.770442163325035183607263531976, −18.895122905244347943321441749774, −18.37601059651843983691447352964, −17.19987525892713430647096670242, −16.48113948938932077152657828466, −16.03448300307487093184295266078, −14.44831637828559341445328529221, −13.61571437804501153164826917861, −12.36600474888201353580576775989, −11.37420115107748664717048580096, −10.30282649204222813218818796402, −9.58882531192305288732438177766, −8.11250650898948395007034966285, −7.26970323099735153927866780976, −6.33806123465386558457809836795, −5.635563494074773163229228018593, −4.08588013410465699498974937844, −2.06551520229285675499152554494, −0.73665531557745548622824922782, 0.96623093423362030673724791765, 2.84462841825438907357748116374, 3.72970909250513521801030085677, 5.29577749161086339942291747683, 6.340936356416545270122581282682, 7.5704440342260404464332707072, 8.91441521184222427529836997094, 9.72773872532122672438801505889, 10.500348980810899878186160096223, 11.491149000135388669402231113693, 12.35873228781197843818968410048, 13.12582002659810144930315519702, 14.95741306635116640870049582961, 16.06755159378783048666536070122, 16.38302004832361561614090099893, 17.78404450286734800077417051606, 18.12688048081422992699761909938, 19.37896505488588278200529235583, 20.20478033036462334522320005434, 21.28554489577554742665462866649, 21.98424908858744254070136948888, 22.692875842100864035966214620045, 23.81375574391377771445147763187, 25.25957122225176993582784832095, 25.836587163247311547662727373229

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