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Properties

Label 1-275-275.214-r0-0-0
Degree $1$
Conductor $275$
Sign $-0.650 + 0.759i$
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  + (−0.309 − 0.951i)2-s + (−0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + 6-s + (−0.309 + 0.951i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)12-s + (0.809 + 0.587i)13-s + 14-s + (0.309 − 0.951i)16-s − 17-s + (−0.309 + 0.951i)18-s + (−0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + 6-s + (−0.309 + 0.951i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)12-s + (0.809 + 0.587i)13-s + 14-s + (0.309 − 0.951i)16-s − 17-s + (−0.309 + 0.951i)18-s + (−0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.650 + 0.759i$
Analytic conductor: \(1.27709\)
Root analytic conductor: \(1.27709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ -0.650 + 0.759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1591375265 + 0.3458162251i\)
\(L(\frac12)\) \(\approx\) \(0.1591375265 + 0.3458162251i\)
\(L(1)\) \(\approx\) \(0.5816037954 + 0.08416376556i\)
\(L(1)\) \(\approx\) \(0.5816037954 + 0.08416376556i\)

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 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 - T \)
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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

−25.27395841784769730727498116069, −24.4735395620062617404012675026, −23.610696911811679408956232537512, −22.9751900345921610115620194255, −22.28211998532062431643299509838, −20.524283098539024543601434411827, −19.59948545312987215611275990191, −18.72231643546352751820035260736, −17.882514301748755568373222072655, −17.09660879354325749138245739321, −16.3613877472472843451423888059, −15.24521575028371676149660846049, −14.02665031982476028185589529637, −13.391833809693535017488153135676, −12.558874727482148681457365472494, −10.96828841574875343774602685882, −10.22479686304393698512484824056, −8.66858120727531967652886936118, −7.98667322270224124527964558585, −6.80306116529898979230782172510, −6.343827686349508986074202423175, −5.06983025942193656271752947559, −3.72933800050432980482128845075, −1.7325428434749584321270607131, −0.29310494334669172741998620403, 1.93521188781396948886900242868, 3.1917006058178847987606535651, 4.188348749780966175837358256531, 5.25959663580713558248699171059, 6.49251542444496539233908977771, 8.40822905177838126185980392097, 9.08967047051424110511393682967, 9.85773980099218339788795686817, 11.11408909874043264353977923590, 11.51358306605329114087626721026, 12.712016121815529742497953569518, 13.71508096225157278430524432309, 15.05868203027610413892097990159, 15.88527548866901882428148526000, 16.93312681103605587380909021911, 17.84144597884689179715781327233, 18.76433004748499188278416021663, 19.72553355385662314309081985513, 20.67988060862734405118293685167, 21.53597287764854477441394362238, 22.04578098125257503429520342700, 22.89937627363100089456037038729, 23.98785413761261010072448639245, 25.66876009726267704614509894528, 26.06344887614200297423098155660

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