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Properties

Label 1-605-605.394-r0-0-0
Degree $1$
Conductor $605$
Sign $-0.750 + 0.660i$
Analytic cond. $2.80960$
Root an. cond. $2.80960$
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  + (−0.198 + 0.980i)2-s + (−0.309 + 0.951i)3-s + (−0.921 − 0.389i)4-s + (−0.870 − 0.491i)6-s + (0.985 − 0.170i)7-s + (0.564 − 0.825i)8-s + (−0.809 − 0.587i)9-s + (0.654 − 0.755i)12-s + (−0.974 + 0.226i)13-s + (−0.0285 + 0.999i)14-s + (0.696 + 0.717i)16-s + (0.362 − 0.931i)17-s + (0.736 − 0.676i)18-s + (0.774 − 0.633i)19-s + (−0.142 + 0.989i)21-s + ⋯
L(s)  = 1  + (−0.198 + 0.980i)2-s + (−0.309 + 0.951i)3-s + (−0.921 − 0.389i)4-s + (−0.870 − 0.491i)6-s + (0.985 − 0.170i)7-s + (0.564 − 0.825i)8-s + (−0.809 − 0.587i)9-s + (0.654 − 0.755i)12-s + (−0.974 + 0.226i)13-s + (−0.0285 + 0.999i)14-s + (0.696 + 0.717i)16-s + (0.362 − 0.931i)17-s + (0.736 − 0.676i)18-s + (0.774 − 0.633i)19-s + (−0.142 + 0.989i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.750 + 0.660i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (394, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ -0.750 + 0.660i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3603223025 + 0.9546895931i\)
\(L(\frac12)\) \(\approx\) \(0.3603223025 + 0.9546895931i\)
\(L(1)\) \(\approx\) \(0.6311057390 + 0.5910999067i\)
\(L(1)\) \(\approx\) \(0.6311057390 + 0.5910999067i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.198 + 0.980i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.985 - 0.170i)T \)
13 \( 1 + (-0.974 + 0.226i)T \)
17 \( 1 + (0.362 - 0.931i)T \)
19 \( 1 + (0.774 - 0.633i)T \)
23 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (-0.254 + 0.967i)T \)
31 \( 1 + (0.0855 + 0.996i)T \)
37 \( 1 + (-0.516 + 0.856i)T \)
41 \( 1 + (0.993 - 0.113i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (0.736 + 0.676i)T \)
53 \( 1 + (-0.696 + 0.717i)T \)
59 \( 1 + (0.993 + 0.113i)T \)
61 \( 1 + (0.198 + 0.980i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (-0.998 + 0.0570i)T \)
73 \( 1 + (-0.897 + 0.441i)T \)
79 \( 1 + (0.941 + 0.336i)T \)
83 \( 1 + (0.466 + 0.884i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.610 - 0.791i)T \)
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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

−22.59811794811931594818717814280, −22.00456513565330380817600124502, −20.89542884172489787606853975904, −20.33292945887230770137326136132, −19.1772167774691490174313260716, −18.86836489070153995063280829265, −17.73130758710572259573586425985, −17.43476989092425958255536291545, −16.491067526767469350037266372936, −14.75790675601398370511739182563, −14.24458058927068059221511132314, −13.198158262586167910443883519678, −12.366960821037543638524838761181, −11.84291317369676774686212135988, −10.965222162060820024573587305811, −10.14315332339361831551434841367, −8.94459052809715042865313741061, −7.9537177937559825279587292638, −7.52085094478927144126052055561, −5.91613483746562958809521862383, −5.098214067821345446402938757116, −3.98633510498641417249709645858, −2.54139690519650743854519851553, −1.886370584777856640269204447379, −0.716775529945814648183089346311, 1.1107014071233244511193710050, 3.01483766771260423746770978780, 4.3312719613560887039143382931, 5.03078100260453682419861666272, 5.596522005306218266330302614801, 7.0291363364076225007835424726, 7.67216787951801778556801165464, 8.90883374179209896369563504645, 9.47246811838986567562321486631, 10.42816139639147813936436985635, 11.35254353236031595693942610819, 12.28926069555671803614457183906, 13.8143446266245093611553146460, 14.3318107492805700133142388198, 15.13041129800105641089777351007, 15.93034923631026813265640353740, 16.62620192743342747092513661645, 17.612314299261406228717294242600, 17.842671485100054618837571660957, 19.17157636352325022725787128200, 20.15510419120701397241351258790, 21.07773721514218502697298637485, 21.95634225423654583191084583767, 22.53711751091663741475765058929, 23.559618378667020473810574818478

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