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Properties

Label 2-201-201.44-c1-0-12
Degree $2$
Conductor $201$
Sign $-0.0112 + 0.999i$
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  + (−1.71 − 0.246i)3-s + (1.77 − 0.916i)4-s + (−3.50 − 3.67i)7-s + (2.87 + 0.845i)9-s + (−3.27 + 1.13i)12-s + (0.687 − 3.56i)13-s + (2.32 − 3.25i)16-s + (−1.65 − 1.58i)19-s + (5.10 + 7.16i)21-s + (3.27 − 3.77i)25-s + (−4.72 − 2.15i)27-s + (−9.59 − 3.32i)28-s + (1.88 + 9.78i)31-s + (5.89 − 1.13i)36-s + (5.89 + 10.2i)37-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)3-s + (0.888 − 0.458i)4-s + (−1.32 − 1.38i)7-s + (0.959 + 0.281i)9-s + (−0.945 + 0.327i)12-s + (0.190 − 0.989i)13-s + (0.580 − 0.814i)16-s + (−0.380 − 0.362i)19-s + (1.11 + 1.56i)21-s + (0.654 − 0.755i)25-s + (−0.909 − 0.415i)27-s + (−1.81 − 0.627i)28-s + (0.338 + 1.75i)31-s + (0.981 − 0.189i)36-s + (0.968 + 1.67i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0112 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0112 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.0112 + 0.999i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ -0.0112 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.625563 - 0.632626i\)
\(L(\frac12)\) \(\approx\) \(0.625563 - 0.632626i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 + 0.246i)T \)
67 \( 1 + (4.25 + 6.99i)T \)
good2 \( 1 + (-1.77 + 0.916i)T^{2} \)
5 \( 1 + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (3.50 + 3.67i)T + (-0.333 + 6.99i)T^{2} \)
11 \( 1 + (10.8 + 2.08i)T^{2} \)
13 \( 1 + (-0.687 + 3.56i)T + (-12.0 - 4.83i)T^{2} \)
17 \( 1 + (-9.86 - 13.8i)T^{2} \)
19 \( 1 + (1.65 + 1.58i)T + (0.904 + 18.9i)T^{2} \)
23 \( 1 + (-16.6 + 15.8i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.88 - 9.78i)T + (-28.7 + 11.5i)T^{2} \)
37 \( 1 + (-5.89 - 10.2i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-40.8 + 3.89i)T^{2} \)
43 \( 1 + (-2.24 - 3.50i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 + (-11.0 - 45.6i)T^{2} \)
53 \( 1 + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (8.39 - 58.3i)T^{2} \)
61 \( 1 + (1.34 + 14.0i)T + (-59.8 + 11.5i)T^{2} \)
71 \( 1 + (-41.1 + 57.8i)T^{2} \)
73 \( 1 + (-5.46 + 0.521i)T + (71.6 - 13.8i)T^{2} \)
79 \( 1 + (-15.3 + 5.31i)T + (62.0 - 48.8i)T^{2} \)
83 \( 1 + (27.1 + 78.4i)T^{2} \)
89 \( 1 + (85.3 - 25.0i)T^{2} \)
97 \( 1 + (15.6 - 9.03i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.25562563371375053085478931668, −10.94988891747023389723296394543, −10.47544043716023912995112904416, −9.765517183632814586772534499072, −7.77889330700467249279685256946, −6.67967668158527822893923836763, −6.31966364966239605143223915987, −4.82743314020434528089748936060, −3.18289351593610619328359476413, −0.877883051765307087338705176927, 2.36139561170558451952814342437, 3.89912736968418891019277037501, 5.73927777204320682948916214421, 6.31598778087580856096172475698, 7.29665195858378248767937125114, 8.908062777854100785494688519071, 9.791095357176581758800813977496, 11.01815942184834014046960020863, 11.77717516902606949956360708233, 12.48359449171860091869666916672

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