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Properties

Label 1-455-455.419-r1-0-0
Degree $1$
Conductor $455$
Sign $0.711 - 0.702i$
Analytic cond. $48.8965$
Root an. cond. $48.8965$
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  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)6-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 12-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s − 18-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)6-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 12-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s − 18-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(455\)    =    \(5 \cdot 7 \cdot 13\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(48.8965\)
Root analytic conductor: \(48.8965\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{455} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 455,\ (1:\ ),\ 0.711 - 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.447112971 - 0.5939856990i\)
\(L(\frac12)\) \(\approx\) \(1.447112971 - 0.5939856990i\)
\(L(1)\) \(\approx\) \(0.9872909408 - 0.2577705062i\)
\(L(1)\) \(\approx\) \(0.9872909408 - 0.2577705062i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−23.77837369179374632717187461761, −23.40750157306015943801964438944, −22.11655299304903509005840159420, −21.84338163112659235199491588818, −20.5384345407011957255347481900, −19.29679790115979081551856476707, −18.53968379449606512100054061309, −17.596965042739884626130939935097, −17.03347197994284061350215662479, −16.06514433891121568311113975136, −15.2544405756401103255225748552, −14.10328340460168671172322776400, −13.28875508922569800106553919568, −12.84402294331950368896363682638, −11.623049474714844274562165442865, −10.9529557209808665272361394250, −9.30173822576674914177214812159, −8.248670237356647712979570449425, −7.53139447023894861601158158318, −6.565776273566166186343375751712, −5.75441284830923355045160483893, −4.973837333930221695020934526198, −3.59309646518340975717853389559, −2.37816404165060995319179684054, −0.68025968458380405165961107280, 0.603015686330198885040058581160, 2.171576026026002718115058504794, 3.30656799375693011244289015955, 4.33786741544901321986692695231, 5.0925689785392641457785743937, 5.952179169379875968207320235020, 7.277722043773269597502486607023, 8.9531892482253966841908464802, 9.599938862095002215067077387048, 10.55898405110978686974974228053, 11.140500800483454307267379875927, 12.21602658094226973225287000530, 12.821783457914777696829852210522, 14.14675171444195281474935307515, 14.827765776851377339307952501732, 15.739449697534137195184393083281, 16.60859137128756062114173768129, 17.91631406302800143293714637981, 18.33918851567884921801683310588, 19.68191165553496122331144024986, 20.66951436533243023523811852782, 20.817426483174253206678662307231, 22.08012468200273634444110519459, 22.58516314557431414450039763446, 23.26093254274113862194975959823

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