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Properties

Label 2-201-67.29-c3-0-30
Degree $2$
Conductor $201$
Sign $-0.918 + 0.395i$
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  + (2.46 − 4.27i)2-s + 3·3-s + (−8.16 − 14.1i)4-s + 11.8·5-s + (7.39 − 12.8i)6-s + (−12.5 − 21.6i)7-s − 41.0·8-s + 9·9-s + (29.2 − 50.5i)10-s + (12.1 + 21.1i)11-s + (−24.4 − 42.4i)12-s + (−1.08 + 1.88i)13-s − 123.·14-s + 35.5·15-s + (−35.9 + 62.2i)16-s + (−6.60 + 11.4i)17-s + ⋯
L(s)  = 1  + (0.871 − 1.51i)2-s + 0.577·3-s + (−1.02 − 1.76i)4-s + 1.05·5-s + (0.503 − 0.871i)6-s + (−0.676 − 1.17i)7-s − 1.81·8-s + 0.333·9-s + (0.923 − 1.59i)10-s + (0.334 + 0.578i)11-s + (−0.589 − 1.02i)12-s + (−0.0232 + 0.0402i)13-s − 2.35·14-s + 0.611·15-s + (−0.561 + 0.972i)16-s + (−0.0942 + 0.163i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.918 + 0.395i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ -0.918 + 0.395i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.694383 - 3.37087i\)
\(L(\frac12)\) \(\approx\) \(0.694383 - 3.37087i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-280. + 471. i)T \)
good2 \( 1 + (-2.46 + 4.27i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 11.8T + 125T^{2} \)
7 \( 1 + (12.5 + 21.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-12.1 - 21.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (1.08 - 1.88i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (6.60 - 11.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-34.8 + 60.2i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (52.0 - 90.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (9.33 + 16.1i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-32.5 - 56.4i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-47.4 + 82.1i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-213. - 369. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 - 73.4T + 7.95e4T^{2} \)
47 \( 1 + (-234. - 406. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 205.T + 1.48e5T^{2} \)
59 \( 1 - 743.T + 2.05e5T^{2} \)
61 \( 1 + (-21.6 + 37.4i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (509. + 882. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (587. - 1.01e3i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (199. + 345. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-534. + 925. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 653.T + 7.04e5T^{2} \)
97 \( 1 + (-380. + 658. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.63265088407883814888273224914, −10.54088138748749088182210891257, −9.818385994242470638740582790846, −9.305800865044739758487596383974, −7.35683100806715695890428818749, −6.03184315755663784114661844831, −4.60954588104278033787052848660, −3.61582965514096358003490655539, −2.41879681583548502440623046058, −1.17212687094281556995501183584, 2.50152732689435524102951488104, 3.89426914078166668519664750299, 5.52532840463255720854252496991, 5.98102126406160088303106594262, 7.02047153097895776112327243508, 8.364771182149143346874335786919, 9.048531882774592104933009351569, 10.07121590218811852572554989630, 12.00510621190836425119907764158, 12.85914319383697284631673982806

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