↑

Properties

Label 1-201-201.38-r0-0-0
Degree $1$
Conductor $201$
Sign $0.667 - 0.744i$
Analytic cond. $0.933440$
Root an. cond. $0.933440$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s − 14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + 22-s + (0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + 5-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s − 14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + 22-s + (0.5 + 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.667 - 0.744i$
Analytic conductor: \(0.933440\)
Root analytic conductor: \(0.933440\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 201,\ (0:\ ),\ 0.667 - 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9918405109 - 0.4430925505i\)
\(L(\frac12)\) \(\approx\) \(0.9918405109 - 0.4430925505i\)
\(L(1)\) \(\approx\) \(0.9336203582 - 0.3301708182i\)
\(L(1)\) \(\approx\) \(0.9336203582 - 0.3301708182i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 - T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 - T \)
97 \( 1 + (0.5 + 0.866i)T \)
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

−27.02940286303471304330996800548, −25.82358656926389789389695862493, −25.09696283144944826869910719996, −24.59312763460282809990125933522, −23.40865345505124499337590350603, −22.390210777962538555980048149029, −21.34554121072412380458260304558, −20.454060448136488610597101251283, −18.74320323203995077752474022035, −18.42818326131780701419097534856, −17.4417656631822072973039442379, −16.45135700651516941767048513702, −15.520166568683593918701945385064, −14.44273263539082661512020186352, −13.70529587386447843818085638097, −12.50534820911257483573174531387, −10.847271938641881414509458941079, −10.06514484897500007413352047362, −8.75745626127968438094199047168, −8.24397829032922856373412186445, −6.68394896742233734265596902473, −5.64837654195269131693597309080, −5.070019872234252937032540573074, −2.86199110011378302199711506165, −1.30778593420364258839666006775, 1.36486160649657063215603750309, 2.32170986484092135523574102479, 3.92765783182629398074497446764, 4.99360420871330041164708195733, 6.69423936791532681645757274338, 7.85517246545294134328142779447, 9.0550096296264703679653182891, 10.04688889553246478430035799846, 10.745753273419052522209101332704, 11.86616814368905981653227679835, 13.22551985360388364365335992061, 13.64156039516041419297502223791, 15.00004370549042274942204728930, 16.68650281062165391667315985665, 17.34581175591814367342021994104, 18.05472411414764072776292042814, 19.16512491990718950873065869221, 20.18547990530526760605121553209, 21.196632214504786973982679117383, 21.45687226580794538226159399618, 22.9171323596654999082767615749, 23.74284271060071593815900922321, 25.22524133507573112092819128871, 26.08386233135809302644009312418, 26.61525095068550146763507109129

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