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Properties

Label 2-201-67.66-c2-0-9
Degree $2$
Conductor $201$
Sign $0.281 - 0.959i$
Analytic cond. $5.47685$
Root an. cond. $2.34026$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.952i·2-s − 1.73i·3-s + 3.09·4-s + 5.09i·5-s + 1.65·6-s + 5.68i·7-s + 6.75i·8-s − 2.99·9-s − 4.85·10-s − 14.2i·11-s − 5.35i·12-s + 23.0i·13-s − 5.41·14-s + 8.82·15-s + 5.92·16-s − 18.5·17-s + ⋯
L(s)  = 1  + 0.476i·2-s − 0.577i·3-s + 0.772·4-s + 1.01i·5-s + 0.275·6-s + 0.811i·7-s + 0.844i·8-s − 0.333·9-s − 0.485·10-s − 1.29i·11-s − 0.446i·12-s + 1.76i·13-s − 0.386·14-s + 0.588·15-s + 0.370·16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.281 - 0.959i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ 0.281 - 0.959i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.42448 + 1.06715i\)
\(L(\frac12)\) \(\approx\) \(1.42448 + 1.06715i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
67 \( 1 + (64.2 + 18.8i)T \)
good2 \( 1 - 0.952iT - 4T^{2} \)
5 \( 1 - 5.09iT - 25T^{2} \)
7 \( 1 - 5.68iT - 49T^{2} \)
11 \( 1 + 14.2iT - 121T^{2} \)
13 \( 1 - 23.0iT - 169T^{2} \)
17 \( 1 + 18.5T + 289T^{2} \)
19 \( 1 - 31.3T + 361T^{2} \)
23 \( 1 - 2.67T + 529T^{2} \)
29 \( 1 - 17.2T + 841T^{2} \)
31 \( 1 + 5.03iT - 961T^{2} \)
37 \( 1 + 35.1T + 1.36e3T^{2} \)
41 \( 1 - 10.8iT - 1.68e3T^{2} \)
43 \( 1 + 62.7iT - 1.84e3T^{2} \)
47 \( 1 + 4.95T + 2.20e3T^{2} \)
53 \( 1 + 77.1iT - 2.80e3T^{2} \)
59 \( 1 + 36.4T + 3.48e3T^{2} \)
61 \( 1 + 85.7iT - 3.72e3T^{2} \)
71 \( 1 - 110.T + 5.04e3T^{2} \)
73 \( 1 - 70.5T + 5.32e3T^{2} \)
79 \( 1 + 28.1iT - 6.24e3T^{2} \)
83 \( 1 - 32.0T + 6.88e3T^{2} \)
89 \( 1 + 130.T + 7.92e3T^{2} \)
97 \( 1 - 73.8iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.09299723501261654420042641426, −11.47727164925769151470461083977, −10.85424367754004116482685173839, −9.220964039170548245021636093291, −8.249030113114746007724114534007, −6.93244455404768534909719046114, −6.55983031735140533491444457546, −5.42975466063326091360989425037, −3.20502778221406896246180657029, −2.07034558766485985896858222477, 1.09057668819336211264840253135, 2.95067827735841003220829680621, 4.35755257748739937146478230817, 5.41171651707071578165964088464, 7.00912312993579838330632771313, 7.948800349353617815563722397238, 9.354069530822287845099604512522, 10.20173458361483494867915649607, 10.88643328673484748580964623141, 12.09888260863149020475778564879

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