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Properties

Label 1-275-275.271-r1-0-0
Degree $1$
Conductor $275$
Sign $0.724 - 0.689i$
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  + (−0.309 + 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (0.809 + 0.587i)6-s + (0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)12-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)17-s + (0.809 − 0.587i)18-s + (0.809 − 0.587i)19-s + (−0.309 − 0.951i)21-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s + (0.809 + 0.587i)6-s + (0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)12-s + (0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)17-s + (0.809 − 0.587i)18-s + (0.809 − 0.587i)19-s + (−0.309 − 0.951i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.724 - 0.689i$
Analytic conductor: \(29.5528\)
Root analytic conductor: \(29.5528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (1:\ ),\ 0.724 - 0.689i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.724201268 - 0.6893516894i\)
\(L(\frac12)\) \(\approx\) \(1.724201268 - 0.6893516894i\)
\(L(1)\) \(\approx\) \(1.110557011 - 0.06204701974i\)
\(L(1)\) \(\approx\) \(1.110557011 - 0.06204701974i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−25.686266096924678220148986471690, −25.043915550780676420522507093960, −23.44180330066484292252098042101, −22.48990234704274568807621869540, −21.72859317819248440218934662367, −20.75879709525740422317720099531, −20.52371366875087871244282323655, −19.27688593223347498795031141918, −18.309045805564954314817743544775, −17.50604592935662849765906929123, −16.345557091523396721404784317955, −15.40960653382443311410675941698, −14.243156638943331834219497239788, −13.54404661706035210304723148183, −11.97112248094167575649898854463, −11.46376257823714725086419933101, −10.26477377637788904654370656158, −9.65581705298499058252096907338, −8.4074552755458639910726901597, −7.96382592702448753400306122068, −5.65886799425349136502954570949, −4.7653462937719544377365296295, −3.58262515949999806502639437328, −2.66962472290847315452623645727, −1.23282758560443356438258152406, 0.73112693141367545406360721218, 1.72103349301422181526354387385, 3.64891773594115517148227490685, 4.98717776726940775308144765094, 6.21812729943999700294419520459, 7.04930006989823258861606100065, 8.05407623644821958712498796892, 8.60572433017785953005980795444, 9.94021697636513321454308728569, 11.17652644138323586913998356437, 12.37706758455463918620449886410, 13.706988933925940304421865417926, 14.02167843239369217664997870704, 15.01680821516102473956003802268, 16.238030900545577511620534813, 17.181824053643865674652422054106, 17.99954971695386897371804446205, 18.64268991213431643300592732131, 19.66301746157028494591066972522, 20.56742573278203918981774727989, 21.86690923994132905079819676660, 23.34221644680682631144752740868, 23.5548331300269470568140550981, 24.49134022170928731124576618644, 25.2304381842329421686319435846

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