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Properties

Label 1-920-920.419-r0-0-0
Degree $1$
Conductor $920$
Sign $0.555 - 0.831i$
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  + (0.142 + 0.989i)3-s + (0.654 − 0.755i)7-s + (−0.959 + 0.281i)9-s + (−0.841 + 0.540i)11-s + (−0.654 − 0.755i)13-s + (0.415 − 0.909i)17-s + (−0.415 − 0.909i)19-s + (0.841 + 0.540i)21-s + (−0.415 − 0.909i)27-s + (−0.415 + 0.909i)29-s + (0.142 − 0.989i)31-s + (−0.654 − 0.755i)33-s + (0.959 − 0.281i)37-s + (0.654 − 0.755i)39-s + (−0.959 − 0.281i)41-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)3-s + (0.654 − 0.755i)7-s + (−0.959 + 0.281i)9-s + (−0.841 + 0.540i)11-s + (−0.654 − 0.755i)13-s + (0.415 − 0.909i)17-s + (−0.415 − 0.909i)19-s + (0.841 + 0.540i)21-s + (−0.415 − 0.909i)27-s + (−0.415 + 0.909i)29-s + (0.142 − 0.989i)31-s + (−0.654 − 0.755i)33-s + (0.959 − 0.281i)37-s + (0.654 − 0.755i)39-s + (−0.959 − 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.555 - 0.831i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9315073971 - 0.4981624903i\)
\(L(\frac12)\) \(\approx\) \(0.9315073971 - 0.4981624903i\)
\(L(1)\) \(\approx\) \(0.9739505031 + 0.04467728055i\)
\(L(1)\) \(\approx\) \(0.9739505031 + 0.04467728055i\)

Euler product

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

−21.74269512421563099384341550184, −21.36445351929606743296964086453, −20.39039152571576014350774053772, −19.42232755160897565723527018693, −18.70224060042928893314366595115, −18.370914102141379839455002236639, −17.26727633425486083420067690565, −16.71290309841233937511040957410, −15.480864779193716743947936413169, −14.664013342932261544803393526858, −14.06831347681028852075994731985, −13.08957805868791418278651889239, −12.3468980948448022690800830837, −11.70542381612253340855851832711, −10.83551497563887008042821994081, −9.72906621106296337603439587300, −8.556005456660961361287216182985, −8.14018466169235877196774436697, −7.278258622962859302898592168308, −6.12067400908428878730697617877, −5.58969849749256122072960242076, −4.41645743276015063887317774496, −3.05777015370089554929124702630, −2.193358179183048715072741546436, −1.36713268726665094656068677631, 0.44565072262963460429825134003, 2.20758973772131464624320518569, 3.06834387117538098914078597547, 4.20240764120584783886202110205, 4.94903136998233064141545883686, 5.54051386828914910200063332160, 7.13129201057583764009770182099, 7.74275530834555575421260718623, 8.71258102625991436465761319163, 9.711329133023334586662933333985, 10.36471581464326022142059259280, 11.02004760324701629101935307526, 11.91736179329111674848518773093, 13.07948205853819260437347448870, 13.83545379353516946723149703170, 14.81744483473222411280148585888, 15.23278699209146309436633249677, 16.18723501779010477614914670166, 17.00180466650777622505686929565, 17.6275214003788701729288306787, 18.502302927723898087532831947892, 19.728015942522614323943098028390, 20.42659844872678172191717543610, 20.74070351231755506029313375683, 21.769714379961414770418358852151

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