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Properties

Label 1-275-275.202-r1-0-0
Degree $1$
Conductor $275$
Sign $-0.163 - 0.986i$
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  + i·2-s + (0.951 − 0.309i)3-s − 4-s + (0.309 + 0.951i)6-s + (−0.587 − 0.809i)7-s − i·8-s + (0.809 − 0.587i)9-s + (−0.951 + 0.309i)12-s + (0.587 + 0.809i)13-s + (0.809 − 0.587i)14-s + 16-s + (−0.951 + 0.309i)17-s + (0.587 + 0.809i)18-s − 19-s + (−0.809 − 0.587i)21-s + ⋯
L(s)  = 1  + i·2-s + (0.951 − 0.309i)3-s − 4-s + (0.309 + 0.951i)6-s + (−0.587 − 0.809i)7-s − i·8-s + (0.809 − 0.587i)9-s + (−0.951 + 0.309i)12-s + (0.587 + 0.809i)13-s + (0.809 − 0.587i)14-s + 16-s + (−0.951 + 0.309i)17-s + (0.587 + 0.809i)18-s − 19-s + (−0.809 − 0.587i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.163 - 0.986i$
Analytic conductor: \(29.5528\)
Root analytic conductor: \(29.5528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (1:\ ),\ -0.163 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5148014053 - 0.6074396576i\)
\(L(\frac12)\) \(\approx\) \(0.5148014053 - 0.6074396576i\)
\(L(1)\) \(\approx\) \(0.9810515149 + 0.1684952662i\)
\(L(1)\) \(\approx\) \(0.9810515149 + 0.1684952662i\)

Euler product

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

−25.81359864925011509092645761950, −25.07325331271664082852686260143, −23.82551284426044685225870374295, −22.57182147872708217908101920078, −21.90750818359771579061449026949, −21.10256260532337716561160644401, −20.12974436639897574383648221792, −19.58268310113254092949557157476, −18.58432134864760376208881540352, −17.90830932006717065641549191847, −16.340924868555654343261199363617, −15.31496908945198947853606848155, −14.5044848542905070626556018432, −13.17187226378230123993245789352, −12.924078256162681309512262363605, −11.512804992223064652519332525739, −10.47280730469754743352281521655, −9.52422344348501640492270745223, −8.78217416980119897949519510727, −7.91736021507994539642787347773, −6.13873697577831765477458099566, −4.77870375912156040305101673641, −3.63759643184669328208643979151, −2.762598349354052018490623728870, −1.750177512897861428491605622060, 0.20414143909031378711322529691, 1.92839631392208060515847662542, 3.74359683700675080358846600951, 4.255045577880981766649069273436, 6.131426423345046060334099190898, 6.87251699135004261218939352689, 7.80363080710721006013771441629, 8.82799485451871009516395582786, 9.55103133791574441730821710973, 10.77495597689542027108458812833, 12.62068450202625000915835945248, 13.32088014648045651256189693306, 14.0656913106131338228050015506, 14.93884182199503992730632000717, 15.92248896359405617404783714000, 16.733464559964208579693524296353, 17.83196216106399234753583619764, 18.80260068328331395081565751846, 19.53912632652900359117135701211, 20.55148800381198040371204418515, 21.68765218015691808245135229810, 22.706201358022144286406486976604, 23.86350837068295782161196195373, 24.12756095735164809560780712322, 25.44082668755375213887749785294

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