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Properties

Label 1-920-920.307-r0-0-0
Degree $1$
Conductor $920$
Sign $0.397 + 0.917i$
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  + (0.755 + 0.654i)3-s + (0.909 − 0.415i)7-s + (0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (0.909 + 0.415i)13-s + (0.540 + 0.841i)17-s + (−0.841 − 0.540i)19-s + (0.959 + 0.281i)21-s + (−0.540 + 0.841i)27-s + (0.841 − 0.540i)29-s + (0.654 + 0.755i)31-s + (−0.909 − 0.415i)33-s + (−0.989 + 0.142i)37-s + (0.415 + 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)3-s + (0.909 − 0.415i)7-s + (0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (0.909 + 0.415i)13-s + (0.540 + 0.841i)17-s + (−0.841 − 0.540i)19-s + (0.959 + 0.281i)21-s + (−0.540 + 0.841i)27-s + (0.841 − 0.540i)29-s + (0.654 + 0.755i)31-s + (−0.909 − 0.415i)33-s + (−0.989 + 0.142i)37-s + (0.415 + 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.397 + 0.917i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.397 + 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.760547710 + 1.155405637i\)
\(L(\frac12)\) \(\approx\) \(1.760547710 + 1.155405637i\)
\(L(1)\) \(\approx\) \(1.407726949 + 0.4392014845i\)
\(L(1)\) \(\approx\) \(1.407726949 + 0.4392014845i\)

Euler product

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

−21.38154906495619964690869893044, −20.78850285472188828997517066985, −20.45222035003188159630193813539, −19.06704473036263949961370326398, −18.7209243252621168625690073073, −17.95388914720235925187645616506, −17.25268511858847915332155581747, −15.83660105664535178842828710851, −15.44236485408657041835267595944, −14.26045940069563717265930112761, −13.94712846453586965769681624020, −12.870019404573562125505477412238, −12.265454696498479659523824143812, −11.24346569083721187914924249818, −10.41906433595362656974204071655, −9.29468389285825609772590133954, −8.21804051395203203680790033096, −8.137738714341118559523709119528, −6.98404942917805865275200378926, −5.92727884932248568074019362718, −5.083213598305371720330489672715, −3.83416191682369690447612559859, −2.83530997336075004973385211333, −2.01974252671063654203425216071, −0.906039357822287982845232860804, 1.40892545930290641662128348720, 2.3813008041755093814243508814, 3.454064815053214482051619406888, 4.41283727396814207785938595298, 4.995361970534406898245888065889, 6.23824902117532491235231920204, 7.44652752430287569426648663305, 8.27948544608318444060337179281, 8.69528557471761563373404443813, 9.983902518376374748232617006835, 10.580431056418580240508574084049, 11.24275060449484668403500168464, 12.482443192026440577994260760317, 13.48708086226961523042622222331, 14.00792925959583823501236112777, 14.95323066602996014719180727242, 15.50061973176157259855632608412, 16.36503091208609373387302813796, 17.252420890230206017559397485772, 18.07816503516493648842130066930, 19.023230558128218055452245626, 19.73407973895680275975317182536, 20.65712596331812929062131550503, 21.26553445928465158969912896881, 21.51916539807779204599682663950

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