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Properties

Label 2-201-67.37-c3-0-12
Degree $2$
Conductor $201$
Sign $-0.459 + 0.888i$
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  + (−2.81 − 4.87i)2-s + 3·3-s + (−11.8 + 20.5i)4-s − 14.5·5-s + (−8.44 − 14.6i)6-s + (−2.46 + 4.26i)7-s + 88.5·8-s + 9·9-s + (41.0 + 71.1i)10-s + (12.2 − 21.2i)11-s + (−35.5 + 61.6i)12-s + (33.4 + 57.9i)13-s + 27.7·14-s − 43.7·15-s + (−154. − 267. i)16-s + (−21.9 − 38.0i)17-s + ⋯
L(s)  = 1  + (−0.995 − 1.72i)2-s + 0.577·3-s + (−1.48 + 2.56i)4-s − 1.30·5-s + (−0.574 − 0.995i)6-s + (−0.132 + 0.230i)7-s + 3.91·8-s + 0.333·9-s + (1.29 + 2.24i)10-s + (0.336 − 0.582i)11-s + (−0.855 + 1.48i)12-s + (0.713 + 1.23i)13-s + 0.529·14-s − 0.753·15-s + (−2.41 − 4.18i)16-s + (−0.313 − 0.543i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.459 + 0.888i$
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: \((2,\ 201,\ (\ :3/2),\ -0.459 + 0.888i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.434798 - 0.714554i\)
\(L(\frac12)\) \(\approx\) \(0.434798 - 0.714554i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-489. + 246. i)T \)
good2 \( 1 + (2.81 + 4.87i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + 14.5T + 125T^{2} \)
7 \( 1 + (2.46 - 4.26i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-12.2 + 21.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-33.4 - 57.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (21.9 + 38.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (1.64 + 2.85i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-33.7 - 58.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-136. + 235. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-66.9 + 115. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (188. + 326. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (123. - 213. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 - 277.T + 7.95e4T^{2} \)
47 \( 1 + (-132. + 230. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 455.T + 1.48e5T^{2} \)
59 \( 1 - 580.T + 2.05e5T^{2} \)
61 \( 1 + (6.53 + 11.3i)T + (-1.13e5 + 1.96e5i)T^{2} \)
71 \( 1 + (-55.2 + 95.6i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-441. - 764. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-525. + 910. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (172. + 299. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 651.T + 7.04e5T^{2} \)
97 \( 1 + (-515. - 893. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.61171177373768743029555156269, −10.91654451071961651862767818868, −9.615848554674103229331113591705, −8.847621972519589523049369526328, −8.175459129940923901679547445826, −7.15544787523734005750924784385, −4.25871436324913079983956689441, −3.66299700535404236515730836079, −2.35993381038881842591939376727, −0.69957498836839968136477824520, 0.933234795165078975360874316055, 3.89197380451222363969056256944, 5.10622061130820784308729462432, 6.63583609961972043211169246717, 7.33258001811699640970921650092, 8.388934240740908189788083558866, 8.650916563016499036794501050552, 10.11145017798638369780127586585, 10.78235199301447022735466666272, 12.54824299552669255094921407464

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