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Properties

Label 1-275-275.189-r1-0-0
Degree $1$
Conductor $275$
Sign $0.668 - 0.743i$
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  + (−0.809 + 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)6-s + (0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (0.309 − 0.951i)17-s + (0.309 + 0.951i)18-s + (−0.309 − 0.951i)19-s + (0.809 + 0.587i)21-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)6-s + (0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)12-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (0.309 − 0.951i)17-s + (0.309 + 0.951i)18-s + (−0.309 − 0.951i)19-s + (0.809 + 0.587i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.668 - 0.743i$
Analytic conductor: \(29.5528\)
Root analytic conductor: \(29.5528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (1:\ ),\ 0.668 - 0.743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.568176215 - 0.6991760023i\)
\(L(\frac12)\) \(\approx\) \(1.568176215 - 0.6991760023i\)
\(L(1)\) \(\approx\) \(1.042548090 - 0.08198966205i\)
\(L(1)\) \(\approx\) \(1.042548090 - 0.08198966205i\)

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 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−25.97617477237003440803421599687, −25.00904050221734485165099509435, −23.91826723078369067099089461937, −22.613525293293356920095288891061, −21.477138019349832424084089183149, −20.86337154960715093079550547457, −20.19318796058721865597671973843, −19.21216767046070827442019386881, −18.592439581871150236618527987656, −17.07476644975362162646070560258, −16.667090151923653315053459037457, −15.52540463743889606333099582111, −14.3250884810636338103461259859, −13.51450398285569213944070870389, −12.32358419521528945167541680131, −11.04826279982902073748872886331, −10.32258038030923980238662236511, −9.526619009022640532149198934959, −8.35181286951930268348541851823, −7.80756674229576029649517255777, −6.468021597781175778551426547879, −4.33472818660206086406234071064, −3.82075576987680167419050805720, −2.39988583423825234190637063252, −1.269104737246947177043573604874, 0.68010123473788107966586614740, 2.007265452218864227560891460969, 3.06071259664052819468479218493, 5.05950308624096177392948258324, 6.11514888892196780645250287856, 7.26062898196620701353335275821, 8.09211017414525108216166407673, 8.93441702570829147475137766192, 9.69292988173240670533121130231, 11.08385721497221292780766766961, 12.16678731540772331752905826827, 13.39500530966864166507716515493, 14.3588422107507877541491583892, 15.31134316249519561530579563219, 15.808817885842596744042332605950, 17.35894004952685259294660543361, 18.12683164040392075586250772070, 18.710411061218144710758052406703, 19.72137381353038078173022686654, 20.42075120846572774068364278867, 21.52657175401117538572572986235, 22.93776732007953663970014275099, 23.92650544380139822132334503207, 24.696410976098784383981382973202, 25.391595994682597833012167868329

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