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Properties

Label 1-920-920.627-r0-0-0
Degree $1$
Conductor $920$
Sign $-0.134 - 0.990i$
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  + (0.540 − 0.841i)3-s + (−0.281 + 0.959i)7-s + (−0.415 − 0.909i)9-s + (−0.654 + 0.755i)11-s + (−0.281 − 0.959i)13-s + (0.989 − 0.142i)17-s + (0.142 − 0.989i)19-s + (0.654 + 0.755i)21-s + (−0.989 − 0.142i)27-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (0.281 + 0.959i)33-s + (0.909 − 0.415i)37-s + (−0.959 − 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)3-s + (−0.281 + 0.959i)7-s + (−0.415 − 0.909i)9-s + (−0.654 + 0.755i)11-s + (−0.281 − 0.959i)13-s + (0.989 − 0.142i)17-s + (0.142 − 0.989i)19-s + (0.654 + 0.755i)21-s + (−0.989 − 0.142i)27-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (0.281 + 0.959i)33-s + (0.909 − 0.415i)37-s + (−0.959 − 0.281i)39-s + (0.415 − 0.909i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.134 - 0.990i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (627, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ -0.134 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9222160753 - 1.056333724i\)
\(L(\frac12)\) \(\approx\) \(0.9222160753 - 1.056333724i\)
\(L(1)\) \(\approx\) \(1.062523174 - 0.3875442753i\)
\(L(1)\) \(\approx\) \(1.062523174 - 0.3875442753i\)

Euler product

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

−21.89383371572431136091633685584, −21.25121790174790967313099757117, −20.60544435058933224017971616795, −19.8295128516252662230040069256, −19.064612542605754020373916409, −18.34426066905298463392085719115, −16.89307341420045050019180697577, −16.52508632910397313499315050283, −15.966284965627116485478161674877, −14.68161710980727935942438984888, −14.27474185604423669220083356377, −13.47165117296254335661352677877, −12.57837064807007577780946574385, −11.30493746815988355658045850106, −10.68031824864560101636442215977, −9.83850843901202087704766244903, −9.23939305261685956570117794769, −8.03652888904949802763129734966, −7.56767774489068748445068140223, −6.25168270128239150761159320126, −5.28652413107578960167676930641, −4.26855963929166410702245596038, −3.55808489701124160099952158090, −2.7000703310653734901595015632, −1.30819338714971159437071869426, 0.5903863854242893788999038591, 2.1181713869738362644588553653, 2.65717759436336720113672135784, 3.644251441403161673642619608898, 5.18675137912607148868631355383, 5.777776708050835063585391870391, 6.972309427795631968044560505298, 7.64001143163870590027547734478, 8.45378281802108498059927302062, 9.354451940102005474969943211343, 10.08741771564745345569372610743, 11.339563587357582835527547002105, 12.32187964388066615283392382105, 12.712472114499273912479710611814, 13.50981427185253700594130436776, 14.56699306062604255916006144375, 15.21200720438377857664000734795, 15.853660281475752668553704887, 17.15319733997629527285855634576, 17.97005830533402833137337317021, 18.4426379290553508790135039586, 19.30481840445684848457982468021, 19.9912002286169158960246667255, 20.76838411236419800182008594005, 21.58170084800901608515149054875

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