↑

Properties

Label 1-287-287.19-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.374 + 0.927i$
Analytic cond. $1.33282$
Root an. cond. $1.33282$
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  + (−0.994 − 0.104i)2-s + (0.258 + 0.965i)3-s + (0.978 + 0.207i)4-s + (0.743 + 0.669i)5-s + (−0.156 − 0.987i)6-s + (−0.951 − 0.309i)8-s + (−0.866 + 0.5i)9-s + (−0.669 − 0.743i)10-s + (0.998 + 0.0523i)11-s + (0.0523 + 0.998i)12-s + (−0.987 + 0.156i)13-s + (−0.453 + 0.891i)15-s + (0.913 + 0.406i)16-s + (−0.0523 + 0.998i)17-s + (0.913 − 0.406i)18-s + (0.358 + 0.933i)19-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (0.258 + 0.965i)3-s + (0.978 + 0.207i)4-s + (0.743 + 0.669i)5-s + (−0.156 − 0.987i)6-s + (−0.951 − 0.309i)8-s + (−0.866 + 0.5i)9-s + (−0.669 − 0.743i)10-s + (0.998 + 0.0523i)11-s + (0.0523 + 0.998i)12-s + (−0.987 + 0.156i)13-s + (−0.453 + 0.891i)15-s + (0.913 + 0.406i)16-s + (−0.0523 + 0.998i)17-s + (0.913 − 0.406i)18-s + (0.358 + 0.933i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.374 + 0.927i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.374 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5159731902 + 0.7648490409i\)
\(L(\frac12)\) \(\approx\) \(0.5159731902 + 0.7648490409i\)
\(L(1)\) \(\approx\) \(0.7295579147 + 0.4114944288i\)
\(L(1)\) \(\approx\) \(0.7295579147 + 0.4114944288i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.994 - 0.104i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (0.743 + 0.669i)T \)
11 \( 1 + (0.998 + 0.0523i)T \)
13 \( 1 + (-0.987 + 0.156i)T \)
17 \( 1 + (-0.0523 + 0.998i)T \)
19 \( 1 + (0.358 + 0.933i)T \)
23 \( 1 + (0.104 - 0.994i)T \)
29 \( 1 + (-0.891 - 0.453i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (-0.777 + 0.629i)T \)
53 \( 1 + (-0.838 + 0.544i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (0.406 - 0.913i)T \)
67 \( 1 + (0.544 + 0.838i)T \)
71 \( 1 + (0.453 + 0.891i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (0.933 - 0.358i)T \)
97 \( 1 + (0.453 - 0.891i)T \)
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

−25.255197759473152124415930614622, −24.43849435547400546006010887689, −24.13495908454283797520886293381, −22.58222220894534144926144869774, −21.40931930578861825957103587011, −20.20166775801792919664074832084, −19.84515588813107092701003614117, −18.81958263098839362473030867334, −17.80326594313492484314375125708, −17.28482731736893749702371775771, −16.48307940862470460821167632800, −15.11105995596073343724927914311, −14.11026591694769857350637490166, −13.122483558373254329180685422889, −12.02040361954293455621113177197, −11.32539484029113064410112353554, −9.47567584683321634490995250084, −9.382604089287185073474093926583, −8.07428391703628458199447399666, −7.14344828305915567026214910769, −6.264248679559538467584487306460, −5.11224755437023352220142781358, −2.95164204794783810805477742244, −1.889716450155919487880707675975, −0.83017783914537736581185267282, 1.80097367571633047318303847251, 2.86229853198944251811400031696, 4.04101423470052994136066397704, 5.70349318546770508186101577509, 6.64959385028109075694203035091, 7.903184056358832768903414939428, 9.05381836053039758046163232198, 9.77476231337997568114015302015, 10.46160301822694353844583293383, 11.37094073343589550298064850114, 12.53815923462042011926702576788, 14.34920108248458902132751993354, 14.66386686037241150226396157722, 15.828737961314781462346266187781, 17.07148546375330555483069114938, 17.19634395878929647048942517771, 18.613859545938272730554127322138, 19.43667304749731995577830889848, 20.30535906878261021291578955315, 21.269417584949524386876532728772, 21.93904798351408300714666263326, 22.76286134469124068605590287504, 24.54096032517302279893645721226, 25.12930123694810455148602257574, 26.09657017300693702887246380505

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