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Properties

Label 1-1275-1275.566-r0-0-0
Degree $1$
Conductor $1275$
Sign $0.0239 + 0.999i$
Analytic cond. $5.92107$
Root an. cond. $5.92107$
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  + (0.891 + 0.453i)2-s + (0.587 + 0.809i)4-s + (−0.923 + 0.382i)7-s + (0.156 + 0.987i)8-s + (0.760 − 0.649i)11-s + (0.951 + 0.309i)13-s + (−0.996 − 0.0784i)14-s + (−0.309 + 0.951i)16-s + (0.987 − 0.156i)19-s + (0.972 − 0.233i)22-s + (−0.760 + 0.649i)23-s + (0.707 + 0.707i)26-s + (−0.852 − 0.522i)28-s + (0.972 − 0.233i)29-s + (−0.852 + 0.522i)31-s + (−0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (0.891 + 0.453i)2-s + (0.587 + 0.809i)4-s + (−0.923 + 0.382i)7-s + (0.156 + 0.987i)8-s + (0.760 − 0.649i)11-s + (0.951 + 0.309i)13-s + (−0.996 − 0.0784i)14-s + (−0.309 + 0.951i)16-s + (0.987 − 0.156i)19-s + (0.972 − 0.233i)22-s + (−0.760 + 0.649i)23-s + (0.707 + 0.707i)26-s + (−0.852 − 0.522i)28-s + (0.972 − 0.233i)29-s + (−0.852 + 0.522i)31-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0239 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0239 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1275\)    =    \(3 \cdot 5^{2} \cdot 17\)
Sign: $0.0239 + 0.999i$
Analytic conductor: \(5.92107\)
Root analytic conductor: \(5.92107\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1275} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1275,\ (0:\ ),\ 0.0239 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.831484850 + 1.875896680i\)
\(L(\frac12)\) \(\approx\) \(1.831484850 + 1.875896680i\)
\(L(1)\) \(\approx\) \(1.587813034 + 0.7543819997i\)
\(L(1)\) \(\approx\) \(1.587813034 + 0.7543819997i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.891 + 0.453i)T \)
7 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (0.760 - 0.649i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.987 - 0.156i)T \)
23 \( 1 + (-0.760 + 0.649i)T \)
29 \( 1 + (0.972 - 0.233i)T \)
31 \( 1 + (-0.852 + 0.522i)T \)
37 \( 1 + (0.760 + 0.649i)T \)
41 \( 1 + (-0.0784 - 0.996i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (-0.156 + 0.987i)T \)
59 \( 1 + (-0.891 + 0.453i)T \)
61 \( 1 + (0.649 + 0.760i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (0.233 + 0.972i)T \)
73 \( 1 + (0.0784 - 0.996i)T \)
79 \( 1 + (-0.852 - 0.522i)T \)
83 \( 1 + (0.987 - 0.156i)T \)
89 \( 1 + (-0.951 + 0.309i)T \)
97 \( 1 + (-0.972 + 0.233i)T \)
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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

−20.74361935412594441558130443132, −19.95764423379197442197609751082, −19.83999608356434433722672348048, −18.61256729146315558458091453004, −18.03850770518079313254829484346, −16.72193165202687280656795432454, −16.13467115733945583038413721384, −15.42214913265907799900005924298, −14.47182689293341789388161636928, −13.86377227284306903081464590708, −13.02609521891696043216897750092, −12.45764258197991748939663179714, −11.61063411884630144697505597955, −10.81944644535459670187951729415, −9.88885452061821738441723652701, −9.448151199444267863409975421623, −8.10397217527733296264570867642, −6.95986338813754048949616925694, −6.39475450750546540704574148864, −5.58449744489589544243558757608, −4.46671993748907873262469254762, −3.72775236649476702675200340854, −3.03629174798361609168355872460, −1.865297623785904274438950793615, −0.8242651965916708728456129695, 1.31934507667955837302319292834, 2.65353918957764416078045352683, 3.48458129562725745873828529972, 4.06682429917896036089750830046, 5.34674816060244325282338763632, 6.04257960197594777372117344246, 6.63389129642107381214895094091, 7.5542865001272547421016434106, 8.61773893341023962100290757486, 9.23201928264693498606234751467, 10.39543732678333224094731416932, 11.46972913248022547645176248921, 11.95806246109087126943957745373, 12.82206313894302922725735886685, 13.78798361276750924234783116939, 13.97184426128314114308187571931, 15.18752585662437462481888806197, 15.92242257361743842462170354401, 16.29365717962900054121024161225, 17.17998255010737042452495499744, 18.13077216162376738123456694775, 18.9996976459464084491093432001, 19.88294524333473395464533565428, 20.491988794225323550411221146781, 21.63481285607090176389168198021

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