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Properties

Label 1-403-403.328-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.993 - 0.110i$
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  + (0.669 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.993 - 0.110i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.993 - 0.110i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1005829370 + 1.823229035i\)
\(L(\frac12)\) \(\approx\) \(-0.1005829370 + 1.823229035i\)
\(L(1)\) \(\approx\) \(0.8028750513 + 1.259555692i\)
\(L(1)\) \(\approx\) \(0.8028750513 + 1.259555692i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.669 + 0.743i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.913 - 0.406i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (0.669 + 0.743i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
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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

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

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