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Properties

Label 2-201-67.37-c3-0-33
Degree $2$
Conductor $201$
Sign $-0.385 + 0.922i$
Analytic cond. $11.8593$
Root an. cond. $3.44374$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.252 + 0.436i)2-s + 3·3-s + (3.87 − 6.70i)4-s − 15.4·5-s + (0.756 + 1.31i)6-s + (10.1 − 17.5i)7-s + 7.94·8-s + 9·9-s + (−3.90 − 6.76i)10-s + (−17.8 + 30.9i)11-s + (11.6 − 20.1i)12-s + (−9.46 − 16.4i)13-s + 10.2·14-s − 46.4·15-s + (−28.9 − 50.1i)16-s + (−31.8 − 55.1i)17-s + ⋯
L(s)  = 1  + (0.0891 + 0.154i)2-s + 0.577·3-s + (0.484 − 0.838i)4-s − 1.38·5-s + (0.0514 + 0.0891i)6-s + (0.545 − 0.945i)7-s + 0.351·8-s + 0.333·9-s + (−0.123 − 0.213i)10-s + (−0.489 + 0.848i)11-s + (0.279 − 0.484i)12-s + (−0.202 − 0.349i)13-s + 0.194·14-s − 0.799·15-s + (−0.452 − 0.784i)16-s + (−0.454 − 0.787i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.385 + 0.922i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ -0.385 + 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.856640 - 1.28596i\)
\(L(\frac12)\) \(\approx\) \(0.856640 - 1.28596i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
67 \( 1 + (-468. + 285. i)T \)
good2 \( 1 + (-0.252 - 0.436i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + 15.4T + 125T^{2} \)
7 \( 1 + (-10.1 + 17.5i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (17.8 - 30.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (9.46 + 16.4i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (31.8 + 55.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (70.8 + 122. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-6.22 - 10.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-16.5 + 28.6i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (34.6 - 60.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-54.8 - 94.9i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-80.4 + 139. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 - 187.T + 7.95e4T^{2} \)
47 \( 1 + (-303. + 525. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 599.T + 1.48e5T^{2} \)
59 \( 1 - 574.T + 2.05e5T^{2} \)
61 \( 1 + (-154. - 267. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
71 \( 1 + (289. - 501. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-482. - 835. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (347. - 601. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-242. - 420. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 2.01T + 7.04e5T^{2} \)
97 \( 1 + (673. + 1.16e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.43845447418021687131796740695, −10.85097699654332290098540042949, −9.844234086777467327457472140902, −8.494862071299077307251835206126, −7.33543254973670591788160205393, −7.03618342726878741620758197263, −4.97234602435422693705391216549, −4.18164574722046259415724689562, −2.45126961353602740125788515107, −0.58350096501428271557363991813, 2.16791847744149113982812249385, 3.44713747086422028945142951178, 4.36633802625190708593429388014, 6.17167230181227493487500191891, 7.70246645709683053698598525611, 8.130146281935693868946147383530, 8.892203661605506540774603206923, 10.72670603272594661317535208476, 11.42828118955243628254991522086, 12.31851417217151064369374409704

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