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Properties

Label 1-920-920.379-r0-0-0
Degree $1$
Conductor $920$
Sign $0.702 - 0.711i$
Analytic cond. $4.27246$
Root an. cond. $4.27246$
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  + (0.959 + 0.281i)3-s + (0.142 − 0.989i)7-s + (0.841 + 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.142 − 0.989i)13-s + (−0.654 + 0.755i)17-s + (0.654 + 0.755i)19-s + (0.415 − 0.909i)21-s + (0.654 + 0.755i)27-s + (0.654 − 0.755i)29-s + (0.959 − 0.281i)31-s + (−0.142 − 0.989i)33-s + (−0.841 − 0.540i)37-s + (0.142 − 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)3-s + (0.142 − 0.989i)7-s + (0.841 + 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.142 − 0.989i)13-s + (−0.654 + 0.755i)17-s + (0.654 + 0.755i)19-s + (0.415 − 0.909i)21-s + (0.654 + 0.755i)27-s + (0.654 − 0.755i)29-s + (0.959 − 0.281i)31-s + (−0.142 − 0.989i)33-s + (−0.841 − 0.540i)37-s + (0.142 − 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.702 - 0.711i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.702 - 0.711i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.900754986 - 0.7938469757i\)
\(L(\frac12)\) \(\approx\) \(1.900754986 - 0.7938469757i\)
\(L(1)\) \(\approx\) \(1.450169268 - 0.2008888591i\)
\(L(1)\) \(\approx\) \(1.450169268 - 0.2008888591i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (0.142 - 0.989i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (0.959 - 0.281i)T \)
37 \( 1 + (-0.841 - 0.540i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + T \)
53 \( 1 + (0.142 - 0.989i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (-0.959 + 0.281i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (-0.415 + 0.909i)T \)
73 \( 1 + (0.654 + 0.755i)T \)
79 \( 1 + (-0.142 - 0.989i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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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

−21.755937159408908863847805783144, −21.201887315476889737813921278237, −20.30423448054039683502356875549, −19.69864883402566936504255813446, −18.7826917800590446190608174277, −18.18960223825829530227898531087, −17.52744533610120672313703585513, −16.10142703315718614688032087827, −15.52237536329711038427595214165, −14.84508659509274119590958299665, −13.94185426337480377097177862232, −13.319644505836467257209275933777, −12.2272880592842049588127734500, −11.8208959703030526652292440734, −10.50222214408610647412561694804, −9.36799501450319771596003027109, −9.079083703689071359713889633305, −8.07136466476491233307939413910, −7.14440786023144655933660094204, −6.494827914976960097838961368678, −5.03496236521983414687290076754, −4.42005358365514772195488803584, −2.96929842838609303884391515042, −2.417849335664021760231694996184, −1.42602353950249375364887175944, 0.83767342705110187246719690332, 2.1278536487417574797848449671, 3.222078107002759510449519685692, 3.86262280819537654233424400332, 4.86338061310520062199320753166, 5.95899418907367279448491277680, 7.12971841867210947405439073653, 8.03924546764772972194829215616, 8.40461603372720201582147983033, 9.65044779352714215095000732274, 10.400846259284458438565442256438, 10.92395866221922579806439017799, 12.26372140109492782391991145868, 13.28455748105903611605927939208, 13.7323793125743568788990303819, 14.49992620864812833129365233217, 15.46632287337491629642219183033, 16.021603092357769835448339451882, 17.02454727814966612739667045932, 17.80136460461770881000669712011, 18.86287977947096570631114037367, 19.51694100440236908899950439496, 20.25891946272200850556755037168, 20.86555435909788876052776802333, 21.59386762336503992417096637342

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