↑

Properties

Label 1-10e2-100.63-r0-0-0
Degree $1$
Conductor $100$
Sign $0.425 + 0.904i$
Analytic cond. $0.464398$
Root an. cond. $0.464398$
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  + (0.587 + 0.809i)3-s − i·7-s + (−0.309 + 0.951i)9-s + (−0.309 − 0.951i)11-s + (0.951 + 0.309i)13-s + (−0.587 + 0.809i)17-s + (−0.809 − 0.587i)19-s + (−0.809 + 0.587i)21-s + (0.951 − 0.309i)23-s + (−0.951 + 0.309i)27-s + (0.809 − 0.587i)29-s + (0.809 + 0.587i)31-s + (0.587 − 0.809i)33-s + (−0.951 − 0.309i)37-s + (0.309 + 0.951i)39-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)3-s − i·7-s + (−0.309 + 0.951i)9-s + (−0.309 − 0.951i)11-s + (0.951 + 0.309i)13-s + (−0.587 + 0.809i)17-s + (−0.809 − 0.587i)19-s + (−0.809 + 0.587i)21-s + (0.951 − 0.309i)23-s + (−0.951 + 0.309i)27-s + (0.809 − 0.587i)29-s + (0.809 + 0.587i)31-s + (0.587 − 0.809i)33-s + (−0.951 − 0.309i)37-s + (0.309 + 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.425 + 0.904i$
Analytic conductor: \(0.464398\)
Root analytic conductor: \(0.464398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 100,\ (0:\ ),\ 0.425 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9945252018 + 0.6311448850i\)
\(L(\frac12)\) \(\approx\) \(0.9945252018 + 0.6311448850i\)
\(L(1)\) \(\approx\) \(1.116965167 + 0.4113014413i\)
\(L(1)\) \(\approx\) \(1.116965167 + 0.4113014413i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.951 - 0.309i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.587 - 0.809i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.587 + 0.809i)T \)
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

−29.832900266768813539034257711680, −29.02018384578455923276437368670, −27.65294207460979484784066573674, −26.43882222609579719114201471778, −25.59938670430543376575185156074, −24.69543251457298384007298432488, −23.353264299525707380585593195249, −22.97324105110013362210526954680, −20.945811001347378095684041394862, −20.32387887200159186321133784531, −19.27108363709598539116334817331, −18.10937713733076699907133975502, −17.26690547100191439525245243594, −15.74162662378227791084954850144, −14.52621042918832994064214819404, −13.46355711731038195812812127609, −12.70464140403861060176890012050, −11.21939014038266345950741975891, −9.888639003003479412881757581506, −8.489993877272501323009328338120, −7.39633804327965915488430324046, −6.425810703736440169816117976168, −4.475976160153197778372340822069, −2.98410970323721462349261115813, −1.34867547335343612410304478024, 2.323827694017852473815328687709, 3.61010660602096720920508715451, 5.05556461014858999001318651088, 6.372505071926371222241753510, 8.52288534515483407010158719891, 8.772269643842701744198712947479, 10.42592513389010368002039980541, 11.36117924760869609443738317533, 12.990296391428464606209369242651, 14.08005584533068657696870549940, 15.35092085432207074743579797057, 15.91000843901192083815359423309, 17.274736384800885513090390438552, 18.80134619550598543721281790129, 19.496644029062362117430060513509, 21.10067834332063788600381611349, 21.42454082407070938031486570647, 22.61653148050436344654674739782, 24.03613814297844306132975612615, 25.149626675453035944987505204665, 26.044038671150277299527457970939, 26.96033710137051902574459784523, 28.084793244518938355504643594897, 28.79254188393758299026699538777, 30.442396680572865243323744729439

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