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Properties

Label 1-403-403.87-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.991 - 0.132i$
Analytic cond. $1.87152$
Root an. cond. $1.87152$
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 + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.991 - 0.132i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.991 - 0.132i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05568596388 - 0.8366934352i\)
\(L(\frac12)\) \(\approx\) \(0.05568596388 - 0.8366934352i\)
\(L(1)\) \(\approx\) \(0.6411193979 - 0.5573606799i\)
\(L(1)\) \(\approx\) \(0.6411193979 - 0.5573606799i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
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

−24.979766418911173440158372993926, −24.04594113549158038772489490906, −23.34934463834045350070816858430, −22.10671923092043711323532840284, −21.54867597206898350638246807145, −20.01193185104214210774716423919, −19.29032136731573769708087647409, −18.66952464410210130082975628806, −18.15156943439889995905758578481, −16.65812678356496225236202882508, −15.681253358056456469542280160957, −15.255507936234137304189216832770, −14.4338492794190852975917270873, −13.57286484588837398487917853066, −12.53934980106132897698480048216, −11.00169979327934668738647541639, −10.15515459052506172347490712304, −9.13877545720669137068713706053, −8.30510402883941760098480183093, −7.64608938101248080112173410075, −6.51794059130452897359458120657, −5.71642593854478008381222202334, −4.09210730073381455468208394291, −3.09783034683316308976243353488, −1.867170826984338916605709186132, 0.49924387815638677653364523526, 1.94382612398608273061021982054, 2.970327619485758504128266395523, 4.21062654742131316287217611592, 4.61953301074493228861632335167, 6.996675035014457279683007795889, 7.73314736524347117287636112632, 8.62202238333396599018372936515, 9.50420846342787966019968768532, 10.162647587925998661242167960986, 11.34253579462782884273088334826, 12.52963965275088239723161827683, 13.12060251117533338661114190609, 13.79350486804121158140256906420, 15.22271829131302274279359248676, 16.11726995996864981033066077206, 16.97945498978475353562100158743, 18.04103361403314243815629051945, 19.00497149298679512379891232491, 19.873977015856588719459291397258, 20.323617079668470734175665779999, 20.8039949602436274820978378022, 21.98469978444668300018114382802, 23.04235103231522576031813162451, 23.9244286877025047453028849777

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