↑

Properties

Label 1-403-403.178-r1-0-0
Degree $1$
Conductor $403$
Sign $-0.979 - 0.201i$
Analytic cond. $43.3083$
Root an. cond. $43.3083$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.979 - 0.201i$
Analytic conductor: \(43.3083\)
Root analytic conductor: \(43.3083\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (1:\ ),\ -0.979 - 0.201i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03124287472 + 0.3070076447i\)
\(L(\frac12)\) \(\approx\) \(0.03124287472 + 0.3070076447i\)
\(L(1)\) \(\approx\) \(0.5959907302 + 0.2324691559i\)
\(L(1)\) \(\approx\) \(0.5959907302 + 0.2324691559i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−23.33008327754312509962339105906, −22.49336614133212433913430252320, −21.841990480314969674514704357779, −21.31645472294812466824322723272, −20.40737763952172023651794718947, −19.241924754816633096678284840202, −18.37789758403356389938895569533, −17.74515828768417873169262601136, −16.77523073390281856538583070712, −16.076524059850168413263838395928, −14.74618141364820622903516733695, −13.393614696974314750499006184741, −12.98570869935245494050490220815, −11.99174248497260522651819795018, −10.87389426309414147322233869729, −10.31924744798611734638725985495, −9.47620901018364957608607690836, −8.69469836874782113447392158774, −6.90421478098786356905532452132, −5.73062015322129168215129252131, −5.19158352497756472730686225829, −3.739849394687899799358038533121, −2.79481596241504686818159960394, −1.382632805470570639198332168036, −0.11662047496213342066576269023, 1.11916401882380562227293641693, 2.760434460774902368482674136655, 4.657780920048784338528417660, 5.28412659525794027190793181081, 6.43387317834153635405352521057, 6.82240656691398790550061405258, 7.89996830407025391701181342844, 9.51433636549621085687920151650, 9.81927573834122808523850169247, 11.01065177100901060078653989753, 12.48901955561422422790108985321, 13.28734573995665854817722271274, 13.66806454197636203455517613545, 15.15122624651895233622541031119, 16.01252209885314902219761206329, 16.76927906136454112568778819272, 17.54613606635941928073127394841, 18.202915878002825768335005250949, 18.90217806089174638469667345546, 20.201141052037175835477552558528, 21.53244381641270075405710088661, 22.411533776123812597005605938781, 22.872209160968902184304192572064, 23.70682000153099927165354104481, 24.8160934920953562671123349750

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