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Properties

Label 1-451-451.193-r0-0-0
Degree $1$
Conductor $451$
Sign $0.359 + 0.933i$
Analytic cond. $2.09443$
Root an. cond. $2.09443$
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  + (−0.951 + 0.309i)2-s + (0.453 − 0.891i)3-s + (0.809 − 0.587i)4-s + (−0.951 + 0.309i)5-s + (−0.156 + 0.987i)6-s + (0.707 + 0.707i)7-s + (−0.587 + 0.809i)8-s + (−0.587 − 0.809i)9-s + (0.809 − 0.587i)10-s + (−0.156 − 0.987i)12-s + (−0.453 + 0.891i)13-s + (−0.891 − 0.453i)14-s + (−0.156 + 0.987i)15-s + (0.309 − 0.951i)16-s + (0.707 − 0.707i)17-s + (0.809 + 0.587i)18-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.453 − 0.891i)3-s + (0.809 − 0.587i)4-s + (−0.951 + 0.309i)5-s + (−0.156 + 0.987i)6-s + (0.707 + 0.707i)7-s + (−0.587 + 0.809i)8-s + (−0.587 − 0.809i)9-s + (0.809 − 0.587i)10-s + (−0.156 − 0.987i)12-s + (−0.453 + 0.891i)13-s + (−0.891 − 0.453i)14-s + (−0.156 + 0.987i)15-s + (0.309 − 0.951i)16-s + (0.707 − 0.707i)17-s + (0.809 + 0.587i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(451\)    =    \(11 \cdot 41\)
Sign: $0.359 + 0.933i$
Analytic conductor: \(2.09443\)
Root analytic conductor: \(2.09443\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{451} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 451,\ (0:\ ),\ 0.359 + 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5657070454 + 0.3884331151i\)
\(L(\frac12)\) \(\approx\) \(0.5657070454 + 0.3884331151i\)
\(L(1)\) \(\approx\) \(0.6793256553 + 0.07909590605i\)
\(L(1)\) \(\approx\) \(0.6793256553 + 0.07909590605i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 + (0.453 - 0.891i)T \)
5 \( 1 + (-0.951 + 0.309i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-0.453 + 0.891i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
19 \( 1 + (-0.156 + 0.987i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (-0.987 - 0.156i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.951 + 0.309i)T \)
67 \( 1 + (0.987 - 0.156i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (0.987 + 0.156i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (0.453 - 0.891i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
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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

−24.09801548766690067941320089996, −22.85308532543685434765399072973, −21.87940906918515751753627499111, −20.84054715021098579197280162653, −20.34675293828041058024556902031, −19.70409734617404805220819312064, −18.92643822632356718621237922780, −17.642904408854788273822972993499, −16.870831872604437738472057440883, −16.21144261472903265633441759911, −15.17345822647634496583490537402, −14.70051706877200817971539513525, −13.16832429395598309453881696052, −12.12393615328391788191654589724, −11.03383893084606920004581981629, −10.6113563939016401846124817080, −9.580238721407280338143109124951, −8.51108361320116591905234623502, −7.962166393786939746933302170941, −7.164308199488167285861001459471, −5.354881936933263907773022816065, −4.17210888354230412015988588348, −3.445805499833571892276768191715, −2.172505385561811793932229407987, −0.52196645015976891117908512697, 1.349602951235587513076520452184, 2.30519925464927310823570193992, 3.49103411778685847651035516064, 5.21630746633990900894344220842, 6.36295077441001960468523942397, 7.42013721750888391484302283715, 7.836818700449813585124321574638, 8.76298159585696031688726281230, 9.60360901272882797501213067771, 11.04221354794706126030560574752, 11.8476220044236521830003058995, 12.29597747446763907575817315913, 14.11217315052366440522711668517, 14.59896751966464083559334444890, 15.46822607012753972488727214023, 16.42929749439291760174505227252, 17.47129153879655999658241254155, 18.40527991227287115961519446541, 18.86347884185675174968421009200, 19.50456945244790014862200673527, 20.456904564411368514655153326898, 21.25791351565964093695093951073, 22.78035793797118581833486228882, 23.71756636722350934970238301377, 24.22574136011080403238250337359

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