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Properties

Label 1-275-275.21-r1-0-0
Degree $1$
Conductor $275$
Sign $-0.637 - 0.770i$
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 + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)6-s − 7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)12-s + (0.809 − 0.587i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s − 18-s + (−0.309 + 0.951i)19-s + (−0.309 − 0.951i)21-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)6-s − 7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)12-s + (0.809 − 0.587i)13-s + (−0.809 − 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s − 18-s + (−0.309 + 0.951i)19-s + (−0.309 − 0.951i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.637 - 0.770i$
Analytic conductor: \(29.5528\)
Root analytic conductor: \(29.5528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (1:\ ),\ -0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.7026340993 + 1.493173467i\)
\(L(\frac12)\) \(\approx\) \(-0.7026340993 + 1.493173467i\)
\(L(1)\) \(\approx\) \(0.8738050859 + 1.056248903i\)
\(L(1)\) \(\approx\) \(0.8738050859 + 1.056248903i\)

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

−24.751465327106031115096790445751, −23.73357469003054653067004869084, −23.18204930482411636415071726102, −22.25044609297322985835462820985, −21.260956734591364355000122494812, −20.09232265411181190967081444486, −19.62183322696045187145101283964, −18.69002445427267893507135436726, −17.91218234542474414288248523436, −16.31044261122624120994168324253, −15.47348628107639038249751992712, −14.174090153713493675513254365005, −13.54837405790561053588613185204, −12.807339434462221566489091001231, −11.89144380828722510449352616897, −11.00774971431095377525690925703, −9.63135235904368626237824769918, −8.75167077132576982447125040517, −7.022187569767497945297447235065, −6.50596565529883397585487830977, −5.311993319826351004468759374506, −3.76918745085764794081382860983, −2.865782884418263634531781889255, −1.72063839265672449841295789761, −0.33980121285379814069455555288, 2.45253725598904165159981149312, 3.63117024524042903786459482849, 4.19787098053179145224818800807, 5.725386195132103093255723166330, 6.28066950205883321977026170412, 7.89314206785728195632515853240, 8.68060705632566669097410435733, 9.95627766272899226140986889313, 10.88573371707902298273552502248, 12.19771787667112039963417491576, 13.18140382062967143123337763422, 14.008857824210761906208661171956, 15.12677617189139783386156369968, 15.676448377425594001392513474, 16.543129216487477642996656207624, 17.29238184124473691223336580122, 18.81330095694610863124008176074, 20.02621379367046126144272028531, 20.762304924488303613072066397574, 21.64157390498423565818883372423, 22.594961218143660052922014979794, 22.94568119436119427421782795424, 24.2514971273564945226563929852, 25.30266287801435579295237000787, 25.96311867001062770569288067507

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