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Properties

Label 2-201-201.164-c1-0-0
Degree $2$
Conductor $201$
Sign $-0.272 - 0.962i$
Analytic cond. $1.60499$
Root an. cond. $1.26688$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.03 − 1.79i)2-s + (−0.401 + 1.68i)3-s + (−1.15 + 2.00i)4-s − 0.547·5-s + (3.44 − 1.02i)6-s + (−1.46 − 0.843i)7-s + 0.643·8-s + (−2.67 − 1.35i)9-s + (0.567 + 0.983i)10-s + (−2.51 + 4.35i)11-s + (−2.90 − 2.74i)12-s + (−1.60 + 0.927i)13-s + 3.50i·14-s + (0.219 − 0.921i)15-s + (1.64 + 2.84i)16-s + (−2.99 + 1.72i)17-s + ⋯
L(s)  = 1  + (−0.734 − 1.27i)2-s + (−0.231 + 0.972i)3-s + (−0.577 + 1.00i)4-s − 0.244·5-s + (1.40 − 0.419i)6-s + (−0.551 − 0.318i)7-s + 0.227·8-s + (−0.892 − 0.450i)9-s + (0.179 + 0.311i)10-s + (−0.758 + 1.31i)11-s + (−0.839 − 0.793i)12-s + (−0.445 + 0.257i)13-s + 0.935i·14-s + (0.0566 − 0.238i)15-s + (0.410 + 0.710i)16-s + (−0.725 + 0.418i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.272 - 0.962i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ -0.272 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.112059 + 0.148280i\)
\(L(\frac12)\) \(\approx\) \(0.112059 + 0.148280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.401 - 1.68i)T \)
67 \( 1 + (-7.74 - 2.65i)T \)
good2 \( 1 + (1.03 + 1.79i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 0.547T + 5T^{2} \)
7 \( 1 + (1.46 + 0.843i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.51 - 4.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.60 - 0.927i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.99 - 1.72i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.617 + 1.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.29 + 0.745i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0127 + 0.00736i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.03 - 3.48i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.75 + 6.50i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.12 - 3.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 7.55iT - 43T^{2} \)
47 \( 1 + (7.90 + 4.56i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.05T + 53T^{2} \)
59 \( 1 + 2.60iT - 59T^{2} \)
61 \( 1 + (-0.355 + 0.205i)T + (30.5 - 52.8i)T^{2} \)
71 \( 1 + (-7.20 - 4.15i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.80 + 13.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.24 + 0.717i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.74 + 3.89i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.1iT - 89T^{2} \)
97 \( 1 + (10.6 - 6.13i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.35256340922788033167705791295, −11.53898455308777219532243948222, −10.56811142149549116260925879338, −9.985089782942522465688213765242, −9.300685863620620661461996100534, −8.152075650776569706980246922350, −6.61269395098867899185825079046, −4.89187888941796685406445296337, −3.75396650924582006023852503362, −2.39568425107686509510755722815, 0.19525329077071337331817063212, 2.88259650934235185561891578059, 5.36725059348317106297128795688, 6.18179463240534387661323717314, 7.08194638414030829893281883176, 8.079843158566819760867126462128, 8.644440356850157126079324294546, 9.914548977065883782987036133002, 11.26423150144352560280594251015, 12.20077155453047863351659317203

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