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Properties

Label 2-201-201.125-c3-0-2
Degree $2$
Conductor $201$
Sign $-0.431 - 0.902i$
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  + (1.46 − 4.98i)3-s + (−3.32 − 7.27i)4-s + (−16.8 + 26.1i)7-s + (−22.7 − 14.5i)9-s + (−41.1 + 5.91i)12-s + (−14.4 + 2.07i)13-s + (−41.9 + 48.3i)16-s + (8.25 − 5.30i)19-s + (105. + 122. i)21-s + (17.7 + 123. i)25-s + (−106. + 91.8i)27-s + (246. + 35.4i)28-s + (129. + 18.5i)31-s + (−30.7 + 213. i)36-s − 416.·37-s + ⋯
L(s)  = 1  + (0.281 − 0.959i)3-s + (−0.415 − 0.909i)4-s + (−0.909 + 1.41i)7-s + (−0.841 − 0.540i)9-s + (−0.989 + 0.142i)12-s + (−0.308 + 0.0443i)13-s + (−0.654 + 0.755i)16-s + (0.0996 − 0.0640i)19-s + (1.10 + 1.27i)21-s + (0.142 + 0.989i)25-s + (−0.755 + 0.654i)27-s + (1.66 + 0.239i)28-s + (0.749 + 0.107i)31-s + (−0.142 + 0.989i)36-s − 1.85·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.431 - 0.902i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ -0.431 - 0.902i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0461309 + 0.0732173i\)
\(L(\frac12)\) \(\approx\) \(0.0461309 + 0.0732173i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.46 + 4.98i)T \)
67 \( 1 + (-261. + 482. i)T \)
good2 \( 1 + (3.32 + 7.27i)T^{2} \)
5 \( 1 + (-17.7 - 123. i)T^{2} \)
7 \( 1 + (16.8 - 26.1i)T + (-142. - 312. i)T^{2} \)
11 \( 1 + (-189. - 1.31e3i)T^{2} \)
13 \( 1 + (14.4 - 2.07i)T + (2.10e3 - 618. i)T^{2} \)
17 \( 1 + (3.21e3 + 3.71e3i)T^{2} \)
19 \( 1 + (-8.25 + 5.30i)T + (2.84e3 - 6.23e3i)T^{2} \)
23 \( 1 + (-1.02e4 - 6.57e3i)T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (-129. - 18.5i)T + (2.85e4 + 8.39e3i)T^{2} \)
37 \( 1 + 416.T + 5.06e4T^{2} \)
41 \( 1 + (-4.51e4 - 5.20e4i)T^{2} \)
43 \( 1 + (439. + 200. i)T + (5.20e4 + 6.00e4i)T^{2} \)
47 \( 1 + (-8.73e4 - 5.61e4i)T^{2} \)
53 \( 1 + (-9.74e4 + 1.12e5i)T^{2} \)
59 \( 1 + (1.97e5 + 5.78e4i)T^{2} \)
61 \( 1 + (639. - 554. i)T + (3.23e4 - 2.24e5i)T^{2} \)
71 \( 1 + (2.34e5 - 2.70e5i)T^{2} \)
73 \( 1 + (325. + 375. i)T + (-5.53e4 + 3.85e5i)T^{2} \)
79 \( 1 + (1.01e3 - 146. i)T + (4.73e5 - 1.38e5i)T^{2} \)
83 \( 1 + (8.13e4 + 5.65e5i)T^{2} \)
89 \( 1 + (-5.93e5 + 3.81e5i)T^{2} \)
97 \( 1 + 941. iT - 9.12e5T^{2} \)
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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

−12.39484139397389639437571477104, −11.59954507356739009422793464847, −10.16379528192111492410700759211, −9.160466435517117209946462844613, −8.579205405022214558296880311616, −7.00611809952662454100574782644, −6.07432666680120546191465619740, −5.21958137781274389767285818009, −3.12195519167493566028951815642, −1.77620702362142540897702498093, 0.03476846108436118780732045018, 3.02631909110696115634589341950, 3.89415466586236948372466548196, 4.83193894260286156873893038790, 6.64365143720286876247627843541, 7.75370833838176937033738964481, 8.738913577452089653366751454759, 9.843085347350903106703275805198, 10.39796461257267471151352661270, 11.66076337854907598973201426915

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