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Properties

Label 1-485-485.142-r0-0-0
Degree $1$
Conductor $485$
Sign $0.585 - 0.810i$
Analytic cond. $2.25233$
Root an. cond. $2.25233$
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  + (0.382 + 0.923i)2-s + (−0.923 − 0.382i)3-s + (−0.707 + 0.707i)4-s − i·6-s + (0.195 − 0.980i)7-s + (−0.923 − 0.382i)8-s + (0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (0.923 − 0.382i)12-s + (−0.980 + 0.195i)13-s + (0.980 − 0.195i)14-s − i·16-s + (0.980 − 0.195i)17-s + (−0.382 + 0.923i)18-s + (−0.980 + 0.195i)19-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (−0.923 − 0.382i)3-s + (−0.707 + 0.707i)4-s − i·6-s + (0.195 − 0.980i)7-s + (−0.923 − 0.382i)8-s + (0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (0.923 − 0.382i)12-s + (−0.980 + 0.195i)13-s + (0.980 − 0.195i)14-s − i·16-s + (0.980 − 0.195i)17-s + (−0.382 + 0.923i)18-s + (−0.980 + 0.195i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(485\)    =    \(5 \cdot 97\)
Sign: $0.585 - 0.810i$
Analytic conductor: \(2.25233\)
Root analytic conductor: \(2.25233\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{485} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 485,\ (0:\ ),\ 0.585 - 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5252389120 - 0.2685355245i\)
\(L(\frac12)\) \(\approx\) \(0.5252389120 - 0.2685355245i\)
\(L(1)\) \(\approx\) \(0.7236894151 + 0.1514382914i\)
\(L(1)\) \(\approx\) \(0.7236894151 + 0.1514382914i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
97 \( 1 \)
good2 \( 1 + (0.382 + 0.923i)T \)
3 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 + (0.195 - 0.980i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (-0.980 + 0.195i)T \)
17 \( 1 + (0.980 - 0.195i)T \)
19 \( 1 + (-0.980 + 0.195i)T \)
23 \( 1 + (0.555 - 0.831i)T \)
29 \( 1 + (-0.831 - 0.555i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (0.831 - 0.555i)T \)
41 \( 1 + (0.555 - 0.831i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.923 + 0.382i)T \)
59 \( 1 + (0.831 - 0.555i)T \)
61 \( 1 + T \)
67 \( 1 + (0.980 - 0.195i)T \)
71 \( 1 + (0.555 + 0.831i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (-0.195 - 0.980i)T \)
89 \( 1 + (0.382 - 0.923i)T \)
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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

−23.72386475708193312508320149984, −22.86687510141633505981924714615, −21.83412175154573292023501560634, −21.61701561380402953906773840016, −20.86799436245818469569134400541, −19.569189678651380403286820335953, −18.80338037926373699124916088683, −18.10562207871099476769717672268, −17.153843808438302453968045612778, −16.18858971806767084571776238510, −15.028447733013758647888565644918, −14.57292484869259013942616922250, −12.97773029494427621422165010173, −12.572975110234063872450236484893, −11.46535775254796912633483282424, −11.07063486705141472811462537094, −9.94303728182413226850792508973, −9.252189301386684686445296059576, −8.05443842702235609654605349978, −6.412661684401453321746185476288, −5.4280876331798930463056790083, −5.03458355066579678565390745076, −3.68149220768503701707270360698, −2.68206050846017937684921016433, −1.31259461166547534044994586691, 0.33970607981518798531742617083, 2.12849612294963727124606159531, 3.9211898073043727158040624875, 4.74734946067938863333055261909, 5.51888965264013947265526702656, 6.72225524531618732117919681541, 7.32218663864742299874724043160, 8.0030927064934397311104661813, 9.59066661473809288516567718745, 10.41156194323706610623625596094, 11.57831003251214407228124549975, 12.67327645058599788632947887451, 12.99324500195550214154859706800, 14.29826061619502043655529555936, 14.90415106171361689242447066147, 16.113667772146374047165089284550, 17.0291148502803043109418755660, 17.161707854875451698768686106119, 18.27784791152791625558747794633, 19.05324448467622620449718240919, 20.43002593028701124079074500338, 21.329785786756918884280187273970, 22.297922240139830329586062312, 23.07812888955069302362697098906, 23.4965399545372631743924622122

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