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Properties

Label 1-201-201.131-r1-0-0
Degree $1$
Conductor $201$
Sign $0.206 - 0.978i$
Analytic cond. $21.6004$
Root an. cond. $21.6004$
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  + (−0.841 − 0.540i)2-s + (0.415 + 0.909i)4-s + (0.654 − 0.755i)5-s + (0.841 + 0.540i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)10-s + (0.654 − 0.755i)11-s + (−0.142 − 0.989i)13-s + (−0.415 − 0.909i)14-s + (−0.654 + 0.755i)16-s + (−0.415 + 0.909i)17-s + (0.841 − 0.540i)19-s + (0.959 + 0.281i)20-s + (−0.959 + 0.281i)22-s + (0.959 + 0.281i)23-s + ⋯
L(s)  = 1  + (−0.841 − 0.540i)2-s + (0.415 + 0.909i)4-s + (0.654 − 0.755i)5-s + (0.841 + 0.540i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)10-s + (0.654 − 0.755i)11-s + (−0.142 − 0.989i)13-s + (−0.415 − 0.909i)14-s + (−0.654 + 0.755i)16-s + (−0.415 + 0.909i)17-s + (0.841 − 0.540i)19-s + (0.959 + 0.281i)20-s + (−0.959 + 0.281i)22-s + (0.959 + 0.281i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.206 - 0.978i$
Analytic conductor: \(21.6004\)
Root analytic conductor: \(21.6004\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 201,\ (1:\ ),\ 0.206 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.251252733 - 1.014526502i\)
\(L(\frac12)\) \(\approx\) \(1.251252733 - 1.014526502i\)
\(L(1)\) \(\approx\) \(0.9163069849 - 0.3694589882i\)
\(L(1)\) \(\approx\) \(0.9163069849 - 0.3694589882i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (0.841 + 0.540i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (-0.415 + 0.909i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (0.959 + 0.281i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (-0.654 - 0.755i)T \)
79 \( 1 + (-0.142 - 0.989i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + T \)
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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

−26.79139584235165663814946334031, −26.04482907098026450452557196721, −25.01235865098829272583714065752, −24.356157296837546546198280237364, −23.17735917083518996554133548123, −22.29775336685322615460111075908, −20.89074477590760459563491694821, −20.17731031696701794455480714450, −18.85442558518442532882097996196, −18.17764174899544173524682336095, −17.26838725454406375133945367301, −16.592793857487905565244142275381, −15.10968542507951390540775599156, −14.42658303736794698508561311092, −13.656810064702328759045186461136, −11.67378352412692441490918195875, −10.94214478271723522429100413944, −9.76117228156964464115272125352, −9.09254057900303407124734274155, −7.45065991583612781731954451467, −6.99982673341680718519533328270, −5.70777348402986167638949231483, −4.40230892577804797222208872461, −2.35901732195398109399799306382, −1.271164906245596898452449183771, 0.84549039482581205524580869001, 1.87657896101624303304854787689, 3.27708056062769572391082211674, 4.90405359421030576417874598437, 6.098398938448225418810209537505, 7.69230210973896305599551076829, 8.70168729864453523755797554646, 9.29919347822865410557468886275, 10.63219618644768741824091805508, 11.51831787614300795460179538080, 12.58484947924865636095569486641, 13.47312784797919571133088828201, 14.916897643788863457618295469000, 16.09625598328348078563083180316, 17.21927743056350268739094725541, 17.6657762060668947629476689772, 18.742492284966106527975404689305, 19.88037681024035012472814532308, 20.59844618343276859297284300936, 21.6238590244748210848507265132, 22.09942157681608952598453594880, 23.99647411648541062958936139874, 24.82990119471810695101736487778, 25.36929103081811845694546253958, 26.728010867675172498610911977236

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