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Properties

Label 1-920-920.669-r0-0-0
Degree $1$
Conductor $920$
Sign $-0.684 + 0.728i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.684 + 0.728i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (669, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ -0.684 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3402918906 + 0.7869081826i\)
\(L(\frac12)\) \(\approx\) \(0.3402918906 + 0.7869081826i\)
\(L(1)\) \(\approx\) \(0.7172701756 + 0.3161823947i\)
\(L(1)\) \(\approx\) \(0.7172701756 + 0.3161823947i\)

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

−21.65399201778978756140816578992, −20.841987721832339185540832194498, −20.05777189538174479579448591714, −19.08984915239012951428734656794, −18.26132832632774707353567535074, −17.72274762126471234562459417101, −16.65167252475799409246276465124, −16.39631805426593074330990746989, −15.39651856838968666916856826426, −14.17407986540850327250634688504, −13.528519743401969253263266348, −12.72717787792277283364314608845, −11.83623798492702426728106463300, −11.038672273778003424541475566969, −10.35953646026294348483764836652, −9.66103606253990615429828449858, −8.05136494689100988588594589690, −7.64525576482141535298424874578, −6.63151647227112419484795662781, −5.6495417791990893728878285258, −5.06040312926030342869213396057, −3.89213901251450217216610176684, −2.88653473592512692798010176796, −1.324196285767543594802813905109, −0.48291689475437093551157035011, 1.38242164620235641520976286816, 2.40611465316609948576009554926, 3.74983262458940939329217518814, 4.83232877970868206823586609518, 5.35163404731468863668698322207, 6.377096882744631572650343850188, 7.13698223337944854995652660684, 8.241268879492960813466585374934, 9.42188841172799598889537520769, 9.84808493735841422753256560439, 11.05003494766414635824985310100, 11.6170899037978891877669016401, 12.47968641764022681505121720329, 12.993016869117587340416446498227, 14.48804562067106626029004433670, 15.02147235803011566755018623112, 16.00051406443508859289306985526, 16.47815144131779457391339215124, 17.59788893327935582941788865348, 18.06530674712528458253802156042, 18.821464167453847979740478910484, 19.78025561786993413736654125701, 20.85616931022765015289868102475, 21.61903851397928666527841225768, 21.97119467235695433668868703954

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