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Properties

Label 1-275-275.159-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: $\chi_{275} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ -0.441 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.237426208 + 1.987164301i\)
\(L(\frac12)\) \(\approx\) \(1.237426208 + 1.987164301i\)
\(L(1)\) \(\approx\) \(1.495575397 + 1.194673435i\)
\(L(1)\) \(\approx\) \(1.495575397 + 1.194673435i\)

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

−25.19612266881443356607698330017, −24.27769639415609186145379360846, −23.579887137427108334471729018410, −22.83812124706109065530792307647, −21.5203068759830376812963349902, −20.88233919786777576398839591188, −19.75260785781526531375658966898, −19.46915752100245882210560823524, −18.431612991659730638098094903567, −17.14041839084378336808961005272, −15.86598631827226014636503810675, −14.79143746890227288096342990294, −13.96727844757100738923771236096, −13.38364759963225908618750558357, −12.42205073048866320238900896968, −11.49864897130160803574792709991, −10.18385221280485649952246288593, −9.450539662573559097930261600412, −7.97341245216133299409789935193, −6.900589438775306360018871534378, −6.01618275376915262249317908805, −4.27476371857563535038803361090, −3.64245976926427677814061138030, −2.31991355046456481304040619804, −1.248636738212448817430943970733, 2.58385072144138246075105452785, 3.013314610607855918223101646056, 4.52814315438403344073258961764, 5.27394509831475869998251027378, 6.5591488996319887351812703911, 7.76912096505459755155527104050, 8.66655392990606181942572909959, 9.58958522839832012849489925505, 10.96982880151106647354769316435, 12.1965251005365502122050837456, 13.09179684428548614899780771718, 14.025013669353707500447485428033, 14.94595697444191975979308833312, 15.64875416252570181638996623168, 16.242659391132714751700910034770, 17.57477861065582520036119095184, 18.64936392864073872666528305552, 20.00499441280646566174049775557, 20.54561378022576525824447342999, 21.8347858685367519666403523100, 22.0799374085165700380184142695, 23.123798967407583409585301299353, 24.60664700958296356493493306596, 24.87475957024566010767754063776, 25.816246543963101402356049301363

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