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Properties

Label 2-201-201.38-c1-0-0
Degree $2$
Conductor $201$
Sign $0.396 + 0.917i$
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  + (−1.35 + 2.35i)2-s + (0.494 + 1.65i)3-s + (−2.69 − 4.67i)4-s − 3.45·5-s + (−4.58 − 1.09i)6-s + (0.770 − 0.445i)7-s + 9.22·8-s + (−2.51 + 1.64i)9-s + (4.69 − 8.13i)10-s + (−1.16 − 2.01i)11-s + (6.41 − 6.78i)12-s + (−2.43 − 1.40i)13-s + 2.42i·14-s + (−1.70 − 5.73i)15-s + (−7.14 + 12.3i)16-s + (2.55 + 1.47i)17-s + ⋯
L(s)  = 1  + (−0.961 + 1.66i)2-s + (0.285 + 0.958i)3-s + (−1.34 − 2.33i)4-s − 1.54·5-s + (−1.87 − 0.445i)6-s + (0.291 − 0.168i)7-s + 3.26·8-s + (−0.836 + 0.547i)9-s + (1.48 − 2.57i)10-s + (−0.349 − 0.606i)11-s + (1.85 − 1.95i)12-s + (−0.675 − 0.389i)13-s + 0.646i·14-s + (−0.441 − 1.48i)15-s + (−1.78 + 3.09i)16-s + (0.618 + 0.357i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.396 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.396 + 0.917i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.396 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0924160 - 0.0607249i\)
\(L(\frac12)\) \(\approx\) \(0.0924160 - 0.0607249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.494 - 1.65i)T \)
67 \( 1 + (4.70 - 6.69i)T \)
good2 \( 1 + (1.35 - 2.35i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 3.45T + 5T^{2} \)
7 \( 1 + (-0.770 + 0.445i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.16 + 2.01i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.43 + 1.40i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.55 - 1.47i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.80 - 4.85i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.63 + 3.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.503 - 0.290i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.699 - 0.403i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.66 - 4.61i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.521 + 0.902i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 3.32iT - 43T^{2} \)
47 \( 1 + (-2.03 + 1.17i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.10T + 53T^{2} \)
59 \( 1 - 8.37iT - 59T^{2} \)
61 \( 1 + (11.3 + 6.52i)T + (30.5 + 52.8i)T^{2} \)
71 \( 1 + (6.03 - 3.48i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.36 + 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.80 + 1.61i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.72 - 5.61i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.2iT - 89T^{2} \)
97 \( 1 + (0.616 + 0.356i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.90979506203484880091629089714, −12.12760321547890285701065240908, −10.68641943874719436931570172637, −10.19492761884302618485967501526, −8.869727105207751967659110449517, −7.954624590957535727080234054544, −7.80704819075251616047562102636, −6.12984887636004038778015226782, −4.94525212935328670433062643788, −3.89181812561464686486651993728, 0.12460936594149699501736219971, 2.02828050282545543417735143692, 3.29536798206761121045453653840, 4.53049980249068038710465968523, 7.31374969097873798846457529447, 7.79443150120007485788432729424, 8.680282357904708445969736413512, 9.683581406418510289719300289902, 11.00293141641192068135886396198, 11.75541811586803407824157916045

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