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Properties

Label 1-403-403.119-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.700 + 0.713i$
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  + (0.866 + 0.5i)2-s − 3-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + (−0.866 − 0.5i)6-s + (−0.866 − 0.5i)7-s + i·8-s + 9-s + (0.5 + 0.866i)10-s + (−0.866 + 0.5i)11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)18-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s − 3-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + (−0.866 − 0.5i)6-s + (−0.866 − 0.5i)7-s + i·8-s + 9-s + (0.5 + 0.866i)10-s + (−0.866 + 0.5i)11-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.700 + 0.713i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.700 + 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5344409927 + 1.274118423i\)
\(L(\frac12)\) \(\approx\) \(0.5344409927 + 1.274118423i\)
\(L(1)\) \(\approx\) \(1.021202838 + 0.6852644047i\)
\(L(1)\) \(\approx\) \(1.021202838 + 0.6852644047i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 - T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 - iT \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−23.95885573974933445607425466336, −22.99849015747099411382721175600, −22.007764604296439367510108368969, −21.88572036029308987911832301555, −20.752689301556877779301637996906, −19.98002668598445750260938571090, −18.51756593439430079756433999106, −18.27480276634645404476648444600, −16.78853925899413802878753780648, −16.03938975381474666697829173493, −15.43098095389429349627866584565, −13.80441427659193364527125847945, −13.30746160628128899351153643833, −12.42054673153429059651716014724, −11.751726126486046090400919067739, −10.550926587471667580794223122356, −9.962680362463582517175375693287, −8.91993936388898847195634046273, −6.98691045320522574936242841255, −6.198902572785986784572191593838, −5.335041258516274115373523149885, −4.78773973851918454305670570910, −3.1933304793113249480125056034, −2.14029650927958497184866923339, −0.65950709721500842667117456828, 1.85007807244827532492264011072, 3.183431349209597888794700653384, 4.326996180509227201717282546824, 5.4779575281697401697666286225, 6.12937996503703396817121266415, 6.926128733675829941951930065592, 7.80528234969676486036449598149, 9.67097975032998299284866706332, 10.37128124086247401948172930746, 11.33746255477634221454461809234, 12.4819539445713185633310313644, 13.17447706842353702412100530069, 13.835357581935064970200608358185, 15.11459609469629501035651957165, 15.9006090438361153254697373208, 16.69653714438065543192515973571, 17.587611939467019066050285926815, 18.13181722780714544606605731932, 19.50828080379885253722772072822, 20.76744679929442011348970701085, 21.53765807899255238603092085265, 22.28053306643628955000492183840, 22.93383304393493178324902611140, 23.555700820755320805090079648053, 24.5183573125327864276223298781

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