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Properties

Label 2-201-3.2-c2-0-27
Degree $2$
Conductor $201$
Sign $0.593 + 0.805i$
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  + 0.777i·2-s + (−1.77 − 2.41i)3-s + 3.39·4-s − 2.01i·5-s + (1.87 − 1.38i)6-s + 2.85·7-s + 5.75i·8-s + (−2.66 + 8.59i)9-s + 1.56·10-s − 18.6i·11-s + (−6.04 − 8.19i)12-s + 11.7·13-s + 2.21i·14-s + (−4.86 + 3.58i)15-s + 9.10·16-s − 14.0i·17-s + ⋯
L(s)  = 1  + 0.388i·2-s + (−0.593 − 0.805i)3-s + 0.848·4-s − 0.402i·5-s + (0.313 − 0.230i)6-s + 0.407·7-s + 0.719i·8-s + (−0.296 + 0.955i)9-s + 0.156·10-s − 1.69i·11-s + (−0.503 − 0.683i)12-s + 0.901·13-s + 0.158i·14-s + (−0.324 + 0.238i)15-s + 0.568·16-s − 0.826i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 + 0.805i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.593 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.593 + 0.805i$
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: \((2,\ 201,\ (\ :1),\ 0.593 + 0.805i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.39752 - 0.706283i\)
\(L(\frac12)\) \(\approx\) \(1.39752 - 0.706283i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.77 + 2.41i)T \)
67 \( 1 + 8.18T \)
good2 \( 1 - 0.777iT - 4T^{2} \)
5 \( 1 + 2.01iT - 25T^{2} \)
7 \( 1 - 2.85T + 49T^{2} \)
11 \( 1 + 18.6iT - 121T^{2} \)
13 \( 1 - 11.7T + 169T^{2} \)
17 \( 1 + 14.0iT - 289T^{2} \)
19 \( 1 + 28.1T + 361T^{2} \)
23 \( 1 + 25.4iT - 529T^{2} \)
29 \( 1 + 9.40iT - 841T^{2} \)
31 \( 1 - 54.0T + 961T^{2} \)
37 \( 1 - 4.00T + 1.36e3T^{2} \)
41 \( 1 - 59.6iT - 1.68e3T^{2} \)
43 \( 1 + 73.8T + 1.84e3T^{2} \)
47 \( 1 - 28.6iT - 2.20e3T^{2} \)
53 \( 1 - 3.52iT - 2.80e3T^{2} \)
59 \( 1 - 70.9iT - 3.48e3T^{2} \)
61 \( 1 - 18.4T + 3.72e3T^{2} \)
71 \( 1 - 14.7iT - 5.04e3T^{2} \)
73 \( 1 - 65.4T + 5.32e3T^{2} \)
79 \( 1 + 28.7T + 6.24e3T^{2} \)
83 \( 1 - 27.6iT - 6.88e3T^{2} \)
89 \( 1 - 169. iT - 7.92e3T^{2} \)
97 \( 1 - 122.T + 9.40e3T^{2} \)
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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

−11.90417274503333119112160325452, −11.22708561121545688999585385043, −10.57761584394309242479190663179, −8.437737352372318239765665202336, −8.195309086543695774925645405260, −6.57606588101754335975610451946, −6.17305146485286049516984608189, −4.88707992661379994494572761699, −2.73438210147370504513443918015, −1.03644850287613529722824614527, 1.83508462288861969162606922360, 3.54528273564799125913346137869, 4.72415339326010937459657649925, 6.21151093583771185804348482355, 6.97897200318378794476807907046, 8.459928858154762508810488143008, 9.912322793378000999236699331956, 10.48905467855710116288750569835, 11.26214462181685526024017993283, 12.11123969810706598372303171901

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