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Properties

Label 1-920-920.37-r0-0-0
Degree $1$
Conductor $920$
Sign $0.0350 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0350 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0350 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.0350 + 0.999i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.0350 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5199135233 + 0.5020087539i\)
\(L(\frac12)\) \(\approx\) \(0.5199135233 + 0.5020087539i\)
\(L(1)\) \(\approx\) \(0.7055356401 + 0.1133330635i\)
\(L(1)\) \(\approx\) \(0.7055356401 + 0.1133330635i\)

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

−21.820821910714439476988464430275, −21.177457762831528282120661867022, −19.96144445094698827040385839005, −19.26011673398867817196423045791, −18.65333343118256369607987516224, −17.24470666836452282144254384952, −17.130380574855521795293422726521, −16.312651430081778844017331804266, −15.529037305278662922834156224562, −14.46924100915424787966163888488, −13.62416747191441659758369487327, −12.699465875114388027312709692946, −11.95315253231844833586305567472, −11.240220610021186813713557780694, −10.2546408191770744919885654871, −9.72102827308979448765332447009, −8.70992657838762194583656476852, −7.357294001026072387939982968116, −6.67137077900473748471534686156, −6.0287383918557848090293656791, −4.879063644639848271035492931146, −4.086572558609547899134254028267, −3.14844001615865160031047803605, −1.58201362539647097119859798281, −0.429483327464394120346868877787, 1.097042640791551025138778715897, 2.31078807062907078952526936872, 3.50717583951307755974696514912, 4.591967058617283082236603998883, 5.56309566423566808992696586980, 6.18251199967245524390487848347, 7.10142614953608441310390667657, 7.917321565296216073555100536803, 9.31696856973783493548078645118, 9.853582078403887616572269217347, 10.72514225858632463838546943603, 11.802364242089774546650132386088, 12.44654245590898614535595724680, 12.782459182238550060650604516524, 14.16784456461050867361325800871, 14.98696431744899782796885973864, 15.85581260826558224711883340807, 16.64718510170648182278827474552, 17.21962865908609378718628600170, 18.13509626841825881233674291921, 18.83406717411265874894833922323, 19.54921306966346492551182722075, 20.51866263752120023692010838164, 21.55330360343297969074735822558, 22.2320836587890597768194032204

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