↑

Properties

Label 1-5e3-125.73-r1-0-0
Degree $1$
Conductor $125$
Sign $-0.556 + 0.830i$
Analytic cond. $13.4331$
Root an. cond. $13.4331$
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  + (0.844 − 0.535i)2-s + (−0.684 + 0.728i)3-s + (0.425 − 0.904i)4-s + (−0.187 + 0.982i)6-s + (−0.587 + 0.809i)7-s + (−0.125 − 0.992i)8-s + (−0.0627 − 0.998i)9-s + (0.535 + 0.844i)11-s + (0.368 + 0.929i)12-s + (−0.998 + 0.0627i)13-s + (−0.0627 + 0.998i)14-s + (−0.637 − 0.770i)16-s + (−0.904 + 0.425i)17-s + (−0.587 − 0.809i)18-s + (−0.728 + 0.684i)19-s + ⋯
L(s)  = 1  + (0.844 − 0.535i)2-s + (−0.684 + 0.728i)3-s + (0.425 − 0.904i)4-s + (−0.187 + 0.982i)6-s + (−0.587 + 0.809i)7-s + (−0.125 − 0.992i)8-s + (−0.0627 − 0.998i)9-s + (0.535 + 0.844i)11-s + (0.368 + 0.929i)12-s + (−0.998 + 0.0627i)13-s + (−0.0627 + 0.998i)14-s + (−0.637 − 0.770i)16-s + (−0.904 + 0.425i)17-s + (−0.587 − 0.809i)18-s + (−0.728 + 0.684i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $-0.556 + 0.830i$
Analytic conductor: \(13.4331\)
Root analytic conductor: \(13.4331\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (1:\ ),\ -0.556 + 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4051957693 + 0.7595022006i\)
\(L(\frac12)\) \(\approx\) \(0.4051957693 + 0.7595022006i\)
\(L(1)\) \(\approx\) \(1.003669006 + 0.09661897856i\)
\(L(1)\) \(\approx\) \(1.003669006 + 0.09661897856i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (0.844 - 0.535i)T \)
3 \( 1 + (-0.684 + 0.728i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (0.535 + 0.844i)T \)
13 \( 1 + (-0.998 + 0.0627i)T \)
17 \( 1 + (-0.904 + 0.425i)T \)
19 \( 1 + (-0.728 + 0.684i)T \)
23 \( 1 + (0.248 + 0.968i)T \)
29 \( 1 + (-0.876 + 0.481i)T \)
31 \( 1 + (-0.425 - 0.904i)T \)
37 \( 1 + (-0.770 + 0.637i)T \)
41 \( 1 + (0.968 + 0.248i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (-0.125 + 0.992i)T \)
53 \( 1 + (0.982 - 0.187i)T \)
59 \( 1 + (0.929 - 0.368i)T \)
61 \( 1 + (0.968 - 0.248i)T \)
67 \( 1 + (0.481 - 0.876i)T \)
71 \( 1 + (-0.992 - 0.125i)T \)
73 \( 1 + (0.368 - 0.929i)T \)
79 \( 1 + (-0.728 - 0.684i)T \)
83 \( 1 + (0.684 + 0.728i)T \)
89 \( 1 + (0.929 + 0.368i)T \)
97 \( 1 + (-0.481 - 0.876i)T \)
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

−28.700836986067758163067381659532, −27.07772553892805429580552441965, −26.17837848638106709524560287525, −24.8008141905526986516462366087, −24.30327999317392264132119903104, −23.26230479584874602027566284258, −22.448579105630519706524206620489, −21.71080190958298182259211231579, −20.109562650502352626194928866270, −19.20319730419571535010344236320, −17.63196297484053013742056533693, −16.865901123683348738718477600244, −16.09417471168757750871412705436, −14.57843761847721036291224709288, −13.5075212602384352890917580495, −12.80799848800786639004890048569, −11.66042907516445893750701227609, −10.65232340099161956602736391377, −8.67847637888586154626426030758, −7.15239093876506192303571699272, −6.67212517801539166643277279234, −5.36500770072047007736238218457, −4.09831099964641608463754848998, −2.45948852410366087357662088058, −0.2650286003191934859696990230, 2.03200429306334126029217338328, 3.61209787843150233583926339788, 4.72214068009906590584553285518, 5.81461914228237742143942134480, 6.84333231583851361312802047362, 9.26016957255380263909825986486, 9.982347929684969905980841074105, 11.25330453854051419471068495892, 12.18185522720931602151500548441, 12.959292787618325827270799402301, 14.800404517089213751371162900316, 15.189955372233468353443238301456, 16.43518780134999323189747010968, 17.63404650314887033645099357157, 19.04929555028507299329367726919, 20.03899043229340713139536091986, 21.17056340617730290027093145295, 22.13744983631103338088336851872, 22.51854622766301739161485422304, 23.66406170157585700738844471963, 24.77161504573513414127440355908, 25.935612582921279423821593087342, 27.484935426382882583447238781763, 28.08724080374658219896610047711, 29.08496142529199143930198176905

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