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Properties

Label 1-600-600.419-r0-0-0
Degree $1$
Conductor $600$
Sign $0.968 - 0.248i$
Analytic cond. $2.78638$
Root an. cond. $2.78638$
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  + 7-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)17-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)23-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + (−0.809 − 0.587i)37-s + (0.809 + 0.587i)41-s − 43-s + (−0.309 + 0.951i)47-s + 49-s + (−0.309 + 0.951i)53-s + (0.809 + 0.587i)59-s + ⋯
L(s)  = 1  + 7-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)17-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)23-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + (−0.809 − 0.587i)37-s + (0.809 + 0.587i)41-s − 43-s + (−0.309 + 0.951i)47-s + 49-s + (−0.309 + 0.951i)53-s + (0.809 + 0.587i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(2.78638\)
Root analytic conductor: \(2.78638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 600,\ (0:\ ),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.611086815 - 0.2035275960i\)
\(L(\frac12)\) \(\approx\) \(1.611086815 - 0.2035275960i\)
\(L(1)\) \(\approx\) \(1.232761252 - 0.06469736419i\)
\(L(1)\) \(\approx\) \(1.232761252 - 0.06469736419i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + T \)
11 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (-0.309 - 0.951i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−23.20468863172844057258086005837, −22.18952584411488090804075508500, −21.548177042035951239797963140246, −20.630477278952872266935557119971, −19.87316857105580314604209780586, −19.0629301738426158823609930351, −17.93899929216409878709562332324, −17.45228873046236165660652649491, −16.55211617494814397627478737401, −15.52587313257984114143458798567, −14.55146012785654517726582632640, −14.14490824885451033427128523940, −12.974989694368330509476103050872, −11.81337517004563383290928105728, −11.50683101948207629564264358719, −10.26707705519287870984243230683, −9.30020513002461216052710787971, −8.58634618299018090544702202319, −7.20346313149975552313971334919, −6.93567720229355403794178042525, −5.15888587211045751698408142871, −4.82393439911478942727343980477, −3.49509327394137993550873347847, −2.23814013328972054843628846544, −1.200829371019982282244298476237, 1.041579379803617240024037076707, 2.165153561872964697672354771261, 3.46334186003789482342818014041, 4.45537061876576828658410295142, 5.48330608853214726760001858883, 6.35795497778599344635317707563, 7.65771967090485126439595088887, 8.22352675482291597604489999359, 9.28195505383328452704376081445, 10.30987513041712273796828064542, 11.15466634032722388682009156767, 12.00801293606572973032539730988, 12.815122790229953399454020993868, 14.01718589518151257727299517551, 14.63690148561944940718238975557, 15.31667996010353077130599737553, 16.67784428088935772766652407353, 17.12463065574714049699718847454, 18.0305794514313292020772100477, 18.9969211798459794835980708521, 19.69385101797196005567627056002, 20.73023270460302498996850415877, 21.32647616291161688150523678409, 22.25977856921133593203329831039, 22.98714534241715858224780628286

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