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Properties

Label 1-55-55.42-r1-0-0
Degree $1$
Conductor $55$
Sign $-0.564 + 0.825i$
Analytic cond. $5.91057$
Root an. cond. $5.91057$
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  + (0.587 − 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.809 + 0.587i)6-s + (−0.951 + 0.309i)7-s + (−0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + i·12-s + (−0.587 + 0.809i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.587 − 0.809i)17-s + (0.951 − 0.309i)18-s + (−0.309 + 0.951i)19-s + 21-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 − 0.951i)4-s + (−0.809 + 0.587i)6-s + (−0.951 + 0.309i)7-s + (−0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + i·12-s + (−0.587 + 0.809i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.587 − 0.809i)17-s + (0.951 − 0.309i)18-s + (−0.309 + 0.951i)19-s + 21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.564 + 0.825i$
Analytic conductor: \(5.91057\)
Root analytic conductor: \(5.91057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 55,\ (1:\ ),\ -0.564 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1254910585 - 0.2377097140i\)
\(L(\frac12)\) \(\approx\) \(-0.1254910585 - 0.2377097140i\)
\(L(1)\) \(\approx\) \(0.5582650867 - 0.3984923196i\)
\(L(1)\) \(\approx\) \(0.5582650867 - 0.3984923196i\)

Euler product

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

−33.3972317117448797198139545745, −32.70718243735159304530482095375, −31.75552811596757328050953058809, −30.152812119147199229004060124728, −29.260134265531057025749206594963, −27.83791827839116251211607933539, −26.65764043320441985017168617526, −25.63347559669679697616249595111, −24.20350275064496837053018178431, −23.27507922154331396441390644218, −22.29696371765646265020715622594, −21.55674510208254045064353890301, −19.7937652796844136118714030625, −17.91651108784196760074711814602, −17.03884423646060031788609271694, −15.96169033870562596391335107928, −15.04015282798119753190261595277, −13.22769554528968976462146741937, −12.41291494096905711580183551529, −10.7975397905020183976481794045, −9.263045590572310105215728082898, −7.2971670846987390589662133826, −6.20809192960580390241631249282, −4.954549567633274975566319314464, −3.47920195324239082434695621476, 0.13204660871554240201874048789, 2.22484979521326401344286539893, 4.20717278249979903079374007182, 5.67841290333901737929619234833, 6.83410117195337592878372170065, 9.35586480273254269189154020272, 10.55673934725459087546198490659, 11.88092814471031123403850962666, 12.64870630183631778509731490583, 13.90494174293131186402375154187, 15.582498659854308020908025119403, 16.87370378618845451828382341726, 18.51636005150040820255940795492, 19.17508926191589952050044587885, 20.71039954492191834510690146688, 22.12176769512923517987097070156, 22.61589585360849223119487248353, 23.82380169365259732663634486570, 24.88254139076496771631346194609, 26.80084431971316125282806273990, 28.06951601209275470034209451104, 29.03987039361986206964337786666, 29.531145478872702742700752766, 30.93805773222340918285133620834, 31.96981387715948465838910815384

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