Properties

Label 1-155-155.24-r1-0-0
Degree $1$
Conductor $155$
Sign $-0.631 - 0.775i$
Analytic cond. $16.6570$
Root an. cond. $16.6570$
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  + (0.809 − 0.587i)2-s + (0.913 − 0.406i)3-s + (0.309 − 0.951i)4-s + (0.5 − 0.866i)6-s + (−0.669 − 0.743i)7-s + (−0.309 − 0.951i)8-s + (0.669 − 0.743i)9-s + (0.978 − 0.207i)11-s + (−0.104 − 0.994i)12-s + (−0.104 + 0.994i)13-s + (−0.978 − 0.207i)14-s + (−0.809 − 0.587i)16-s + (−0.978 − 0.207i)17-s + (0.104 − 0.994i)18-s + (−0.104 − 0.994i)19-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.913 − 0.406i)3-s + (0.309 − 0.951i)4-s + (0.5 − 0.866i)6-s + (−0.669 − 0.743i)7-s + (−0.309 − 0.951i)8-s + (0.669 − 0.743i)9-s + (0.978 − 0.207i)11-s + (−0.104 − 0.994i)12-s + (−0.104 + 0.994i)13-s + (−0.978 − 0.207i)14-s + (−0.809 − 0.587i)16-s + (−0.978 − 0.207i)17-s + (0.104 − 0.994i)18-s + (−0.104 − 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $-0.631 - 0.775i$
Analytic conductor: \(16.6570\)
Root analytic conductor: \(16.6570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 155,\ (1:\ ),\ -0.631 - 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.559776596 - 3.283352407i\)
\(L(\frac12)\) \(\approx\) \(1.559776596 - 3.283352407i\)
\(L(1)\) \(\approx\) \(1.642601955 - 1.361706298i\)
\(L(1)\) \(\approx\) \(1.642601955 - 1.361706298i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
7 \( 1 + (-0.669 - 0.743i)T \)
11 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (-0.104 + 0.994i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.104 - 0.994i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 - T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−27.81699126003730903738810978866, −26.80938046111395866176362019356, −25.8667586066808784916739677014, −24.88297861831150747093852428457, −24.70810091484848702508342238767, −22.90469417321718240501671401398, −22.25426059466838537112623271356, −21.38427022977035341361283137171, −20.28000503299195682587209697841, −19.47359848919819921310954199507, −18.045272280189190386858957357573, −16.68460285575611962873320298550, −15.742695279296265438238156778, −14.96725168486832723721542283280, −14.16925824979738817720708508820, −12.94878207415327091928391300768, −12.243467630828801379712013330186, −10.57538067758357799763051248292, −9.136750641594861413248675209247, −8.35064428975416025677196619410, −7.01229739038534398362765146005, −5.8598713437233221162664800335, −4.464739663518267073194160658432, −3.390852564897823918859312337691, −2.31389010794395777601307282813, 0.969856174933726523846832909574, 2.353495421305437313883607863577, 3.596500422398215442192192348712, 4.4618084986518629406192066372, 6.47837360066168094368391910626, 7.05344185261790339681603725767, 8.978324159031640388397870650771, 9.731014129315765447568674389488, 11.175054145146914591334982739900, 12.22513502221693213745471528467, 13.53295147038659307246400104778, 13.7558592786049679334109152843, 14.98842363288209948615070916929, 16.00281659573816542101485847310, 17.50674510650038190760732182642, 19.09449942200830014843180296137, 19.526891031860023577503338098449, 20.32411668298314846912015327366, 21.4433967664149539312352083342, 22.34000573604095787474810963989, 23.5521176323429658899968294562, 24.24910303088584814753753711203, 25.2548556852917695739478940907, 26.32832966630465249890272668663, 27.30344327871249580965585639646

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