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Properties

Label 2-201-67.29-c3-0-3
Degree $2$
Conductor $201$
Sign $0.932 - 0.361i$
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  + (2.19 − 3.79i)2-s − 3·3-s + (−5.60 − 9.71i)4-s − 15.2·5-s + (−6.57 + 11.3i)6-s + (12.5 + 21.7i)7-s − 14.1·8-s + 9·9-s + (−33.5 + 58.0i)10-s + (2.42 + 4.20i)11-s + (16.8 + 29.1i)12-s + (−36.6 + 63.4i)13-s + 109.·14-s + 45.8·15-s + (13.9 − 24.1i)16-s + (−5.49 + 9.51i)17-s + ⋯
L(s)  = 1  + (0.774 − 1.34i)2-s − 0.577·3-s + (−0.701 − 1.21i)4-s − 1.36·5-s + (−0.447 + 0.774i)6-s + (0.676 + 1.17i)7-s − 0.623·8-s + 0.333·9-s + (−1.06 + 1.83i)10-s + (0.0664 + 0.115i)11-s + (0.404 + 0.701i)12-s + (−0.782 + 1.35i)13-s + 2.09·14-s + 0.789·15-s + (0.218 − 0.377i)16-s + (−0.0784 + 0.135i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.932 - 0.361i$
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: \((2,\ 201,\ (\ :3/2),\ 0.932 - 0.361i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.13503 + 0.212200i\)
\(L(\frac12)\) \(\approx\) \(1.13503 + 0.212200i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-297. + 460. i)T \)
good2 \( 1 + (-2.19 + 3.79i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 15.2T + 125T^{2} \)
7 \( 1 + (-12.5 - 21.7i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-2.42 - 4.20i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (36.6 - 63.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (5.49 - 9.51i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-19.0 + 33.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (11.8 - 20.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-30.7 - 53.2i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-157. - 273. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (204. - 353. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-39.5 - 68.5i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 328.T + 7.95e4T^{2} \)
47 \( 1 + (-134. - 233. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 648.T + 1.48e5T^{2} \)
59 \( 1 - 365.T + 2.05e5T^{2} \)
61 \( 1 + (278. - 483. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (498. + 862. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-168. + 291. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-388. - 672. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-276. + 478. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 333.T + 7.04e5T^{2} \)
97 \( 1 + (358. - 620. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.96184037337439322441648157157, −11.56829460722096909148424073512, −10.62679289016039072025360830766, −9.335852526464717106217973978824, −8.123067280038130430780807947830, −6.77823293469373568961192961918, −5.01371030834211535504291787147, −4.52297425211931984773530015591, −3.11019148574148985333405340896, −1.63791200628095881634246481897, 0.43263463700030341285153720342, 3.74549797390491739251351422484, 4.54865281030002369075730423002, 5.52625631340684987666116821165, 6.90068670128061387248907112899, 7.71250091787115342137225779531, 8.099206244893751257535064544405, 10.19073052929096579477183452915, 11.12347387484744585049695539002, 12.11579807549483448416651807918

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