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Properties

Label 2-201-67.29-c3-0-15
Degree $2$
Conductor $201$
Sign $0.939 + 0.343i$
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  + (1.48 − 2.58i)2-s + 3·3-s + (−0.439 − 0.760i)4-s − 11.6·5-s + (4.46 − 7.74i)6-s + (9.08 + 15.7i)7-s + 21.2·8-s + 9·9-s + (−17.4 + 30.1i)10-s + (32.7 + 56.6i)11-s + (−1.31 − 2.28i)12-s + (18.0 − 31.2i)13-s + 54.1·14-s − 35.0·15-s + (35.1 − 60.8i)16-s + (16.7 − 28.9i)17-s + ⋯
L(s)  = 1  + (0.526 − 0.912i)2-s + 0.577·3-s + (−0.0549 − 0.0951i)4-s − 1.04·5-s + (0.304 − 0.526i)6-s + (0.490 + 0.849i)7-s + 0.937·8-s + 0.333·9-s + (−0.550 + 0.953i)10-s + (0.896 + 1.55i)11-s + (−0.0317 − 0.0549i)12-s + (0.384 − 0.665i)13-s + 1.03·14-s − 0.603·15-s + (0.548 − 0.950i)16-s + (0.238 − 0.412i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.939 + 0.343i$
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: \((2,\ 201,\ (\ :3/2),\ 0.939 + 0.343i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.82706 - 0.500584i\)
\(L(\frac12)\) \(\approx\) \(2.82706 - 0.500584i\)
\(L(\frac{5}{2})\) 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 \( 1 + (527. - 149. i)T \)
good2 \( 1 + (-1.48 + 2.58i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 11.6T + 125T^{2} \)
7 \( 1 + (-9.08 - 15.7i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-32.7 - 56.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-18.0 + 31.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-16.7 + 28.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-46.3 + 80.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (53.5 - 92.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-98.1 - 170. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (44.0 + 76.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-33.4 + 58.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-31.4 - 54.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 420.T + 7.95e4T^{2} \)
47 \( 1 + (-57.7 - 100. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 431.T + 1.48e5T^{2} \)
59 \( 1 + 294.T + 2.05e5T^{2} \)
61 \( 1 + (-169. + 293. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (114. + 197. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (85.3 - 147. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (450. + 780. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-497. + 862. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.49e3T + 7.04e5T^{2} \)
97 \( 1 + (243. - 421. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.89276374186354486148072087716, −11.44966064082169720645867504421, −10.11581507492144257054464313941, −9.013257773074229606161341515765, −7.85270239305179702968631569481, −7.12401938629790528303371779750, −5.04285483172768306171713784391, −4.04698677061012872058935244708, −2.97681772707518394284667277244, −1.64049314037178771926700422733, 1.20631888160868661208650802004, 3.69863577311153492882261569387, 4.28447897583591182338587275011, 5.91906119583937566207392834940, 6.92224499172423879780050533718, 7.971752041556337632793824346832, 8.506114326466972274371036962952, 10.17324563787374007697959188950, 11.17797574525152392710401992114, 12.01060042579555188757814348181

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