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Properties

Label 1-920-920.403-r0-0-0
Degree $1$
Conductor $920$
Sign $0.965 + 0.259i$
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  + (0.281 − 0.959i)3-s + (−0.989 − 0.142i)7-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.989 + 0.142i)13-s + (0.755 + 0.654i)17-s + (0.654 + 0.755i)19-s + (−0.415 + 0.909i)21-s + (−0.755 + 0.654i)27-s + (−0.654 + 0.755i)29-s + (0.959 − 0.281i)31-s + (0.989 − 0.142i)33-s + (−0.540 + 0.841i)37-s + (−0.142 + 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯
L(s)  = 1  + (0.281 − 0.959i)3-s + (−0.989 − 0.142i)7-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.989 + 0.142i)13-s + (0.755 + 0.654i)17-s + (0.654 + 0.755i)19-s + (−0.415 + 0.909i)21-s + (−0.755 + 0.654i)27-s + (−0.654 + 0.755i)29-s + (0.959 − 0.281i)31-s + (0.989 − 0.142i)33-s + (−0.540 + 0.841i)37-s + (−0.142 + 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.965 + 0.259i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.965 + 0.259i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.180322804 + 0.1559568647i\)
\(L(\frac12)\) \(\approx\) \(1.180322804 + 0.1559568647i\)
\(L(1)\) \(\approx\) \(0.9878009539 - 0.1365373563i\)
\(L(1)\) \(\approx\) \(0.9878009539 - 0.1365373563i\)

Euler product

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

−21.79375148525641290428256929474, −21.22557332470062119765084317571, −20.2262621343167396834357482991, −19.482411631804752376773485591655, −19.049521009410099796778247126025, −17.77279009759668341341951434517, −16.82009128155122252702604140993, −16.25728416414853697706744156092, −15.60947457327138921856771606406, −14.696844020228781328581011931265, −13.95452903600913125422569986875, −13.17390769504835206098894260985, −12.013246876299245021021155606757, −11.34742415319224545878950697763, −10.2533811253047220021042945899, −9.5967869099215552004849742591, −9.04593531435105658926042292407, −7.97840774599258458173261689602, −6.974785099226948757370269872359, −5.847842343689723832217315563857, −5.147068261861576991838262631726, −4.03312037427022717325291584614, −3.143230025703290309003557445399, −2.536577261070399776776652032983, −0.57297791721277454067446080164, 1.09447823300471301610536984717, 2.148770593823986418809291079136, 3.1201063681485783213254000608, 4.040048462735896370336423031057, 5.41878890863805973016113682693, 6.29277783260674745472454775899, 7.17336668584554714458751498271, 7.656406600133449918214436752569, 8.84512893213599703161548242913, 9.67722888478795112477365110544, 10.35338026638543244074079815377, 11.84995506117387568640960011694, 12.307733529582039506045212611094, 12.93708719712483817470309566967, 13.926753460084915301377066352395, 14.59049168367465112544014147194, 15.402965219076022090447722442926, 16.61388663780389560074186833526, 17.16061767121994219336110529711, 18.01302273409350860684743726601, 19.01435529685952857930820801570, 19.36552932546360948689498315871, 20.20369524603110492628488690724, 20.88352845632638128703651313016, 22.26881160606484570748949210378

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