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Properties

Label 2-201-3.2-c2-0-36
Degree $2$
Conductor $201$
Sign $0.0317 + 0.999i$
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  + 2.78i·2-s + (−0.0951 − 2.99i)3-s − 3.73·4-s − 3.67i·5-s + (8.33 − 0.264i)6-s − 7.35·7-s + 0.738i·8-s + (−8.98 + 0.570i)9-s + 10.2·10-s − 8.95i·11-s + (0.355 + 11.1i)12-s − 17.3·13-s − 20.4i·14-s + (−11.0 + 0.349i)15-s − 16.9·16-s − 29.6i·17-s + ⋯
L(s)  = 1  + 1.39i·2-s + (−0.0317 − 0.999i)3-s − 0.933·4-s − 0.734i·5-s + (1.38 − 0.0441i)6-s − 1.05·7-s + 0.0922i·8-s + (−0.997 + 0.0634i)9-s + 1.02·10-s − 0.813i·11-s + (0.0296 + 0.933i)12-s − 1.33·13-s − 1.46i·14-s + (−0.733 + 0.0232i)15-s − 1.06·16-s − 1.74i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0317 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0317 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.0317 + 0.999i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ 0.0317 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.454085 - 0.439899i\)
\(L(\frac12)\) \(\approx\) \(0.454085 - 0.439899i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0951 + 2.99i)T \)
67 \( 1 + 8.18T \)
good2 \( 1 - 2.78iT - 4T^{2} \)
5 \( 1 + 3.67iT - 25T^{2} \)
7 \( 1 + 7.35T + 49T^{2} \)
11 \( 1 + 8.95iT - 121T^{2} \)
13 \( 1 + 17.3T + 169T^{2} \)
17 \( 1 + 29.6iT - 289T^{2} \)
19 \( 1 - 10.1T + 361T^{2} \)
23 \( 1 + 3.49iT - 529T^{2} \)
29 \( 1 - 23.7iT - 841T^{2} \)
31 \( 1 + 58.8T + 961T^{2} \)
37 \( 1 - 36.6T + 1.36e3T^{2} \)
41 \( 1 - 0.640iT - 1.68e3T^{2} \)
43 \( 1 - 73.4T + 1.84e3T^{2} \)
47 \( 1 + 69.6iT - 2.20e3T^{2} \)
53 \( 1 - 49.4iT - 2.80e3T^{2} \)
59 \( 1 + 38.6iT - 3.48e3T^{2} \)
61 \( 1 + 34.0T + 3.72e3T^{2} \)
71 \( 1 - 47.2iT - 5.04e3T^{2} \)
73 \( 1 + 3.60T + 5.32e3T^{2} \)
79 \( 1 + 129.T + 6.24e3T^{2} \)
83 \( 1 + 30.0iT - 6.88e3T^{2} \)
89 \( 1 - 2.74iT - 7.92e3T^{2} \)
97 \( 1 - 97.5T + 9.40e3T^{2} \)
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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

−12.31832823168167809916748668557, −11.25777493873472464717895997540, −9.418398447277165837353893104104, −8.797286568495631247818095749323, −7.47212032175644500238583044122, −7.05222825441306443773742515706, −5.83495070414530973842378242595, −5.05272325922244066040343188346, −2.79990747615965218356860440655, −0.33733092081525444805569798319, 2.40151767604283514790734392520, 3.41263075614653134504147048636, 4.41040699658505885958850364468, 6.05482071023377289505728785556, 7.38698935991206936935489163615, 9.229745790859404691271428600001, 9.820439695332012174387648390122, 10.47613881084970196005088475711, 11.24241613538131314863568541816, 12.39175318799219747895226665937

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