Properties

Degree 1
Conductor $ 13 \cdot 31 $
Sign $-0.993 + 0.110i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.669 − 0.743i)2-s + (0.309 − 0.951i)3-s + (−0.104 − 0.994i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.978 − 0.207i)10-s + (−0.104 − 0.994i)11-s + (−0.978 − 0.207i)12-s + (0.913 − 0.406i)14-s + (−0.978 + 0.207i)15-s + (−0.978 + 0.207i)16-s + (−0.104 + 0.994i)17-s + (−0.978 + 0.207i)18-s + ⋯
L(s,χ)  = 1  + (0.669 − 0.743i)2-s + (0.309 − 0.951i)3-s + (−0.104 − 0.994i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.913 + 0.406i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.978 − 0.207i)10-s + (−0.104 − 0.994i)11-s + (−0.978 − 0.207i)12-s + (0.913 − 0.406i)14-s + (−0.978 + 0.207i)15-s + (−0.978 + 0.207i)16-s + (−0.104 + 0.994i)17-s + (−0.978 + 0.207i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.993 + 0.110i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.993 + 0.110i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.993 + 0.110i$
motivic weight  =  \(0\)
character  :  $\chi_{403} (360, \cdot )$
Sato-Tate  :  $\mu(15)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 403,\ (0:\ ),\ -0.993 + 0.110i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1005829370 - 1.823229035i$
$L(\frac12,\chi)$  $\approx$  $-0.1005829370 - 1.823229035i$
$L(\chi,1)$  $\approx$  0.8028750513 - 1.259555692i
$L(1,\chi)$  $\approx$  0.8028750513 - 1.259555692i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−24.914758521423723151396447675776, −23.77560066452370182415618875823, −22.875388843665591390326531704623, −22.476448429964913196437390634361, −21.473678538585990219476926656366, −20.56844723666323843811363373400, −20.06350427250390459609740547446, −18.40260899204532563868156709572, −17.72138010185198272357053676290, −16.584659366343570798008374857132, −15.847473598503400357391643605290, −14.96443688328750400298800151086, −14.42358500875572917428041283738, −13.81948553203239876719230362674, −12.285978364660622667421580166443, −11.40619443106824358840442909186, −10.51528442986720588918901955444, −9.399478253017640702180355116798, −8.08157790164319670371819739747, −7.5571253129981793812366087215, −6.44372601950243783056990089729, −4.9968744202294962156398006915, −4.47114290190352963645550335472, −3.42443292020950750744207602270, −2.43388264114663255200158222720, 0.86354636864655839465104813341, 1.78920106117122294892379008814, 2.99412612842176655464143310630, 4.12654314837371211553336868716, 5.314060029488524477366830160402, 6.049700073885586071691229617654, 7.56673111065138817070792391362, 8.50063348408941723625048810779, 9.20012634870158746617964371784, 10.836819278655247765641757171923, 11.74144522418623695669475048467, 12.14260327351608211906852218704, 13.34951757050039167045028649258, 13.71769970788826866196613536369, 14.921126745508294342040306311050, 15.64368737402800245284363447192, 17.097501580290479100469753718567, 18.04136754786598156917555949611, 18.97886136965763623900677329091, 19.63895183562262155230026098266, 20.36734928868546643044371178292, 21.26083496945007293546475478807, 21.97556381254603845442569926986, 23.4333600488209463360781122875, 23.82708578367774880661897171297

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