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Properties

Label 1-920-920.773-r0-0-0
Degree $1$
Conductor $920$
Sign $0.909 + 0.415i$
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  + (0.989 + 0.142i)3-s + (0.755 − 0.654i)7-s + (0.959 + 0.281i)9-s + (0.841 + 0.540i)11-s + (0.755 + 0.654i)13-s + (−0.909 + 0.415i)17-s + (−0.415 + 0.909i)19-s + (0.841 − 0.540i)21-s + (0.909 + 0.415i)27-s + (0.415 + 0.909i)29-s + (−0.142 − 0.989i)31-s + (0.755 + 0.654i)33-s + (−0.281 + 0.959i)37-s + (0.654 + 0.755i)39-s + (−0.959 + 0.281i)41-s + ⋯
L(s)  = 1  + (0.989 + 0.142i)3-s + (0.755 − 0.654i)7-s + (0.959 + 0.281i)9-s + (0.841 + 0.540i)11-s + (0.755 + 0.654i)13-s + (−0.909 + 0.415i)17-s + (−0.415 + 0.909i)19-s + (0.841 − 0.540i)21-s + (0.909 + 0.415i)27-s + (0.415 + 0.909i)29-s + (−0.142 − 0.989i)31-s + (0.755 + 0.654i)33-s + (−0.281 + 0.959i)37-s + (0.654 + 0.755i)39-s + (−0.959 + 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.909 + 0.415i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.909 + 0.415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.453673568 + 0.5340349303i\)
\(L(\frac12)\) \(\approx\) \(2.453673568 + 0.5340349303i\)
\(L(1)\) \(\approx\) \(1.678291229 + 0.1673387975i\)
\(L(1)\) \(\approx\) \(1.678291229 + 0.1673387975i\)

Euler product

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

−21.60809852755896708140141801801, −21.0944335509818548969874703530, −20.13110208476797951444880515739, −19.60324871161769296829540167342, −18.69699896104038878465924938015, −17.99238753787798441245583843462, −17.29132316991352698343925423326, −15.92292846358318389679501504861, −15.426465835102166790184081798, −14.62755327534794660368099033946, −13.833771453746235011647230652, −13.21472240733035528671653331114, −12.19007797871094573903662087843, −11.33139247838246663920069624760, −10.50169128868574824868844124676, −9.20924006912413840467526619118, −8.74947331855516031045044584978, −8.100767254279672452413557936559, −7.0054513849353968347392449921, −6.15667161454159030752993394423, −4.958432611821845494566904271392, −4.02966299478709477250745853533, −3.025137603732684988567803797631, −2.14827611856005224681905232361, −1.10904769492600134167419890655, 1.498092489055245899788761736930, 1.948107716655005415071398915383, 3.49985372207168797787774169069, 4.11175347561078333956817729254, 4.87657182478450747762450325834, 6.4581905357751713862169473098, 7.0784394300524178755487538303, 8.21935508422114365108156280093, 8.65635077809901761875669006530, 9.678427642265366275654269848535, 10.479676240911563048673248871289, 11.362733725702780906224586106595, 12.32826232698590806654440979585, 13.44020407726437934226598411669, 13.8919842159196276936636157317, 14.82479005239155612959293315791, 15.231618166727304708574649210798, 16.48120570406560876768990098747, 17.05271782932932028456801266205, 18.16863945255400316126667986927, 18.77383976333744805354367473669, 19.9420739936852597081977717080, 20.14833456791992272475853011518, 21.09549897043228399484344563037, 21.68303237554841781478686891905

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