Properties

Label 2-201-67.37-c3-0-24
Degree $2$
Conductor $201$
Sign $0.683 + 0.730i$
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  + (0.244 + 0.423i)2-s − 3·3-s + (3.88 − 6.72i)4-s + 20.3·5-s + (−0.732 − 1.26i)6-s + (8.86 − 15.3i)7-s + 7.70·8-s + 9·9-s + (4.97 + 8.62i)10-s + (−16.5 + 28.6i)11-s + (−11.6 + 20.1i)12-s + (−9.19 − 15.9i)13-s + 8.66·14-s − 61.1·15-s + (−29.1 − 50.5i)16-s + (−8.45 − 14.6i)17-s + ⋯
L(s)  = 1  + (0.0863 + 0.149i)2-s − 0.577·3-s + (0.485 − 0.840i)4-s + 1.82·5-s + (−0.0498 − 0.0863i)6-s + (0.478 − 0.829i)7-s + 0.340·8-s + 0.333·9-s + (0.157 + 0.272i)10-s + (−0.452 + 0.783i)11-s + (−0.280 + 0.485i)12-s + (−0.196 − 0.339i)13-s + 0.165·14-s − 1.05·15-s + (−0.455 − 0.789i)16-s + (−0.120 − 0.208i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.683 + 0.730i$
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: \((2,\ 201,\ (\ :3/2),\ 0.683 + 0.730i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.16388 - 0.938860i\)
\(L(\frac12)\) \(\approx\) \(2.16388 - 0.938860i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-68.3 - 544. i)T \)
good2 \( 1 + (-0.244 - 0.423i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 - 20.3T + 125T^{2} \)
7 \( 1 + (-8.86 + 15.3i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (16.5 - 28.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (9.19 + 15.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (8.45 + 14.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (23.6 + 40.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-68.3 - 118. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (22.8 - 39.4i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-77.5 + 134. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-64.6 - 111. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-42.2 + 73.0i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + 505.T + 7.95e4T^{2} \)
47 \( 1 + (-287. + 497. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 40.5T + 1.48e5T^{2} \)
59 \( 1 + 562.T + 2.05e5T^{2} \)
61 \( 1 + (-323. - 559. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
71 \( 1 + (93.4 - 161. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (344. + 596. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (5.63 - 9.76i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-281. - 486. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.34e3T + 7.04e5T^{2} \)
97 \( 1 + (-269. - 465. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.68435858019484917955804117175, −10.55759522812597286007057453652, −10.18760108720787649015417083517, −9.292231153389003615378252717662, −7.37992908508382838630368699042, −6.55754313472942010464617525783, −5.49017421525775309579175794233, −4.84560019337920805745894524336, −2.28907292788638884216178353909, −1.17042021878268972145767918617, 1.77992530971628074847657854879, 2.80947253521944212492320912922, 4.84337960399033559223473180949, 5.91240341561032404115412800904, 6.65235607175816231946217504169, 8.225897351449122619515454448095, 9.145228766068668850015537590133, 10.39206993418924747808369361045, 11.10501850627988053376799542651, 12.23024259710983581878051351032

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