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Properties

Label 2-201-67.38-c2-0-1
Degree $2$
Conductor $201$
Sign $-0.354 - 0.935i$
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  + (−3.01 − 1.73i)2-s − 1.73i·3-s + (4.04 + 7.01i)4-s + 4.44i·5-s + (−3.01 + 5.21i)6-s + (8.21 − 4.74i)7-s − 14.2i·8-s − 2.99·9-s + (7.72 − 13.3i)10-s + (−17.4 + 10.0i)11-s + (12.1 − 7.01i)12-s + (−22.3 − 12.8i)13-s − 32.9·14-s + 7.69·15-s + (−8.57 + 14.8i)16-s + (−4.68 + 8.11i)17-s + ⋯
L(s)  = 1  + (−1.50 − 0.869i)2-s − 0.577i·3-s + (1.01 + 1.75i)4-s + 0.888i·5-s + (−0.501 + 0.869i)6-s + (1.17 − 0.677i)7-s − 1.78i·8-s − 0.333·9-s + (0.772 − 1.33i)10-s + (−1.59 + 0.918i)11-s + (1.01 − 0.584i)12-s + (−1.71 − 0.990i)13-s − 2.35·14-s + 0.513·15-s + (−0.535 + 0.928i)16-s + (−0.275 + 0.477i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.354 - 0.935i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.354 - 0.935i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0299908 + 0.0434446i\)
\(L(\frac12)\) \(\approx\) \(0.0299908 + 0.0434446i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
67 \( 1 + (24.1 - 62.5i)T \)
good2 \( 1 + (3.01 + 1.73i)T + (2 + 3.46i)T^{2} \)
5 \( 1 - 4.44iT - 25T^{2} \)
7 \( 1 + (-8.21 + 4.74i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (17.4 - 10.0i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (22.3 + 12.8i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (4.68 - 8.11i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (5.14 - 8.90i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-8.64 + 14.9i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (13.1 + 22.7i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (49.1 - 28.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (13.4 - 23.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (1.17 - 0.679i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 52.9iT - 1.84e3T^{2} \)
47 \( 1 + (39.5 + 68.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 46.7iT - 2.80e3T^{2} \)
59 \( 1 + 12.2T + 3.48e3T^{2} \)
61 \( 1 + (-14.7 - 8.52i)T + (1.86e3 + 3.22e3i)T^{2} \)
71 \( 1 + (-8.31 - 14.4i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-51.9 + 89.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (45.6 - 26.3i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-31.3 + 54.3i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 54.2T + 7.92e3T^{2} \)
97 \( 1 + (39.1 + 22.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.30864159171349338052299213048, −11.12064462032632055933541993926, −10.49195963977032069549272421218, −9.984833292987934976985976002283, −8.356721734106184955430327545171, −7.59332370159572436178864793686, −7.18743692424247182620980094617, −5.00836936393529363737234909394, −2.84144730161446913830752423610, −1.91266246740641316705574521949, 0.04511610415522494430780155733, 2.13718974906951264084267901164, 5.00040619120877671867612708045, 5.41750696453939428598924175126, 7.24130792443179759940515989433, 8.075862794088600963672959287718, 8.959207768320226004365503545724, 9.449673190567912172394159156012, 10.75913604271784544343179646658, 11.43097747096892940098821129572

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