Properties

Label 1-275-275.141-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: $\chi_{275} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ 0.650 + 0.759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.121904215 + 0.5162772853i\)
\(L(\frac12)\) \(\approx\) \(1.121904215 + 0.5162772853i\)
\(L(1)\) \(\approx\) \(1.139666858 + 0.06548875854i\)
\(L(1)\) \(\approx\) \(1.139666858 + 0.06548875854i\)

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

−25.43316766825774982992807135088, −24.562168386441813286910718315020, −23.85644041556234902671538999194, −23.17854411371282969842104278057, −22.29916445269516982737603853517, −20.9929990734768435772367101875, −20.05188094353894909065965907197, −18.974351898933520975483111426647, −18.06308853669195492185391686028, −17.13121391486956370241574956384, −16.62521599329780510886437841711, −14.915546739146664303667354560183, −14.58377117286508597319958592249, −13.41353382077650446811425238908, −12.85246969905816160044031541311, −11.77752549167558126597398442715, −10.28008310446572545589166724367, −8.96185144008112119758576648680, −7.88639012832816173413135880609, −7.332509081523730449855738757945, −6.35864298130755833379618639292, −5.17790195451456743844688817215, −3.938680167199191869095820589630, −2.63299858362161659519247245809, −0.74250674713473595991863064781, 1.8918832796454049232607512255, 2.90544383540964401755056898758, 4.02086099955503291320702784374, 5.05169686641220846920301297971, 5.8623072248554413343017251737, 7.94031459313919406028927150582, 9.048046107702529750562746031, 9.6837155574902493206271065773, 10.69919959722557268601445204639, 11.6957056862519874076495452538, 12.43595586662515841567043992905, 13.80255153060498426013122543976, 14.68501199519011569476519256624, 15.23792629686533818232105260827, 16.58572680665735670445000484787, 17.62092850711693332233680836966, 18.9426862825543296195670279175, 19.39757085519289275366213812671, 20.65763018479266801136525768489, 21.29626501598227729395687790391, 21.85225943611461085927880522560, 22.765272269212752452856944931611, 23.768324973381979589944268633007, 24.98329485499676894767839980942, 25.92316322649982102051681849871

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