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Properties

Label 1-275-275.123-r0-0-0
Degree $1$
Conductor $275$
Sign $0.668 - 0.743i$
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 + (−0.587 − 0.809i)3-s + (−0.309 + 0.951i)4-s + (−0.309 + 0.951i)6-s + (−0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.309 + 0.951i)9-s + (0.951 − 0.309i)12-s + (−0.951 − 0.309i)13-s + (0.809 + 0.587i)14-s + (−0.809 − 0.587i)16-s + (0.951 + 0.309i)17-s + (0.951 − 0.309i)18-s + (0.309 + 0.951i)19-s + (0.809 + 0.587i)21-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.587 − 0.809i)3-s + (−0.309 + 0.951i)4-s + (−0.309 + 0.951i)6-s + (−0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.309 + 0.951i)9-s + (0.951 − 0.309i)12-s + (−0.951 − 0.309i)13-s + (0.809 + 0.587i)14-s + (−0.809 − 0.587i)16-s + (0.951 + 0.309i)17-s + (0.951 − 0.309i)18-s + (0.309 + 0.951i)19-s + (0.809 + 0.587i)21-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.668 - 0.743i$
Analytic conductor: \(1.27709\)
Root analytic conductor: \(1.27709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ 0.668 - 0.743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5262255863 - 0.2346192336i\)
\(L(\frac12)\) \(\approx\) \(0.5262255863 - 0.2346192336i\)
\(L(1)\) \(\approx\) \(0.5386705083 - 0.2462109034i\)
\(L(1)\) \(\approx\) \(0.5386705083 - 0.2462109034i\)

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 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.951 + 0.309i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 - T \)
43 \( 1 - iT \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.587 + 0.809i)T \)
71 \( 1 + T \)
73 \( 1 - iT \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
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

−25.9820059457030932569373173357, −25.07864295679064001651395705155, −23.92795505119918067964643944446, −23.10363484537401860388164110116, −22.44003974782768796402018330590, −21.44829934055237000213193195240, −20.1297462087416800654468947667, −19.34515205091041217325804124314, −18.26835640532203969593933151864, −17.184037675669327253754757038626, −16.629615804562723604672907271484, −15.85846011301020257497090643200, −14.971189682583227900722383541607, −14.0323412792430800577846527415, −12.69657229965540595815562988948, −11.430021889376448421953880664464, −10.288195436505388612729616496869, −9.68006743491678755361587553243, −8.84371823052108702257190641836, −7.28158424744669484690314905992, −6.547633459138157643573211885045, −5.37433766601417467655872739016, −4.53041668345997367454523355405, −3.02008539171304517725119712666, −0.7180742427489782163447608954, 0.93670518187787454967545077679, 2.34323104219065567983838790004, 3.36683166609744084348612087522, 5.029671137471249480673851704549, 6.284171117231753996007794998608, 7.420897326700270679191118098717, 8.26220788015816518162026042067, 9.67531090697496618195273518356, 10.31740416059504496868326672838, 11.63755609996839190346550155088, 12.3132539842330719347457843493, 12.970033993031611470290529532800, 14.02500512071496525683727122747, 15.64827482084437426313456476957, 16.90952533344756294067090688621, 17.20352488394132420865544877468, 18.59524485946317334074399692954, 18.98783196470603323118935179842, 19.78147659042187375082302445161, 20.93367582641677322816420129700, 22.07120423219784628302363125996, 22.6497360829537649171931515249, 23.54918180778533167937372470387, 25.06735333617859187986953180844, 25.33052314427337538567159652750

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