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Properties

Label 1-385-385.27-r0-0-0
Degree $1$
Conductor $385$
Sign $0.732 - 0.680i$
Analytic cond. $1.78793$
Root an. cond. $1.78793$
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)8-s + (0.809 − 0.587i)9-s − i·12-s + (−0.587 − 0.809i)13-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 − i·23-s + (−0.809 + 0.587i)24-s + (−0.309 + 0.951i)26-s + (−0.587 + 0.809i)27-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)8-s + (0.809 − 0.587i)9-s − i·12-s + (−0.587 − 0.809i)13-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 − i·23-s + (−0.809 + 0.587i)24-s + (−0.309 + 0.951i)26-s + (−0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $0.732 - 0.680i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ 0.732 - 0.680i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5631901332 - 0.2213036816i\)
\(L(\frac12)\) \(\approx\) \(0.5631901332 - 0.2213036816i\)
\(L(1)\) \(\approx\) \(0.5702316687 - 0.1478745733i\)
\(L(1)\) \(\approx\) \(0.5702316687 - 0.1478745733i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
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

−24.52306981337685886740823957773, −23.89259866803490576964687916644, −23.08056854310876833343045483537, −22.27794838147899973603578734248, −21.36133447078782794498206307309, −19.833847034603159130024179048633, −19.16886632485622927409989504101, −18.14982045890744751995213823785, −17.59039944563137861763925205443, −16.744915932533713326565861462460, −15.9756330064906299295762947154, −15.17294818780629443615580347032, −13.8993243705645624581542598181, −13.1874257880157492191166407987, −11.73162550160886600905012799233, −11.16177233135308425537578213397, −9.90122892241057493798277821835, −9.224655258015859994217777477312, −7.827911393715647597401278403983, −7.03136102705715136527818789107, −6.28647058791022751577570640099, −5.16227267709564319109817015920, −4.43826254078787968712409965836, −2.23970286026160318318634760568, −0.84753137993017220708832164753, 0.750291382994239346816908483132, 2.17448897800149745504862130953, 3.585790916445460227291701575317, 4.54504004319968657127290522240, 5.700438597508741767777476036116, 6.94073663416981217218503411437, 8.03252749350735284998280025126, 9.130351361948597491576278831392, 10.30139537636663045901350264621, 10.571827480109992228639618703244, 11.790877269084877507213767149687, 12.43062620384749789663076275977, 13.25554057230820364372724086927, 14.744444986705761172292293772878, 15.83082793503513656123809386839, 16.81466613593914964057763152733, 17.363322754130050695976472525550, 18.27325254373284400024188164312, 18.992232863136028471211019654842, 20.20917056603984444081568535746, 20.78580617784318318468820856838, 22.02303794595380619758158586589, 22.249082097281595429436205283746, 23.28712913433323842326275193445, 24.37960533436481737992178479133

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