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Properties

Label 1-275-275.16-r0-0-0
Degree $1$
Conductor $275$
Sign $0.441 - 0.897i$
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 + (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

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.441 - 0.897i$
Analytic conductor: \(1.27709\)
Root analytic conductor: \(1.27709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ 0.441 - 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5694303958 - 0.3545897514i\)
\(L(\frac12)\) \(\approx\) \(0.5694303958 - 0.3545897514i\)
\(L(1)\) \(\approx\) \(0.5876604988 - 0.2515391835i\)
\(L(1)\) \(\approx\) \(0.5876604988 - 0.2515391835i\)

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

−25.90806963175863282705814695801, −24.86585070161625106025756894398, −24.35283291155889190591658699645, −22.82811853817357570928372550965, −22.72505669622803714272407348408, −21.09418490133750792527417886407, −20.56938305251473511934133128666, −19.07651880618592830963247049358, −18.2456723942006193623294593560, −17.630184869073982638841355326032, −16.60270780796828456250780027848, −15.76724086025793167422466246953, −15.16550752831414889706760237812, −14.110835148725879007261815544553, −12.39066567341592687628810735573, −11.54117524997883095842094726544, −10.52351381042003592871880342957, −9.713777103322415053472179254, −8.70477691344521032276346551895, −7.68073174626692654770348187080, −6.27681495473232515021660144195, −5.60939639679414175442743527486, −4.64803297985388783186233817995, −2.76794397034716863066506222515, −0.9817889733127275429563889637, 0.95602430063393293527792139722, 1.92993686285261541352894968246, 3.62679593192140060325343237157, 4.8400746884216364246013709622, 6.48881018427576283130971525233, 7.25129266482319666931283378554, 8.19579977155611583660824494773, 9.47235375592876060434671440395, 10.666618038408067073662614877212, 11.18143686130575419755085278770, 12.15245406650520253664253188944, 13.1429786904888421200462219303, 14.008677741577048129826752753153, 15.79137325386955055601653444529, 16.692241066637066926652722011389, 17.420515429310042862633175592234, 18.033402250697638261050202493523, 19.24501850773460441161455813838, 19.66035860629223064887287804537, 21.02910697484761190725812481068, 21.67454727980513501398475388372, 22.84640309241487185528552056822, 23.75947520447939730274834949507, 24.511636955733644477365576292433, 25.811783660482524524144877704277

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