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Properties

Label 2-201-67.37-c3-0-10
Degree $2$
Conductor $201$
Sign $0.865 + 0.501i$
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  + (−1.83 − 3.17i)2-s − 3·3-s + (−2.70 + 4.69i)4-s + 12.9·5-s + (5.49 + 9.51i)6-s + (1.64 − 2.84i)7-s − 9.46·8-s + 9·9-s + (−23.7 − 41.1i)10-s + (−15.4 + 26.8i)11-s + (8.12 − 14.0i)12-s + (34.7 + 60.2i)13-s − 12.0·14-s − 38.9·15-s + (38.9 + 67.5i)16-s + (41.6 + 72.1i)17-s + ⋯
L(s)  = 1  + (−0.647 − 1.12i)2-s − 0.577·3-s + (−0.338 + 0.586i)4-s + 1.16·5-s + (0.373 + 0.647i)6-s + (0.0886 − 0.153i)7-s − 0.418·8-s + 0.333·9-s + (−0.751 − 1.30i)10-s + (−0.424 + 0.735i)11-s + (0.195 − 0.338i)12-s + (0.741 + 1.28i)13-s − 0.229·14-s − 0.669·15-s + (0.609 + 1.05i)16-s + (0.594 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.865 + 0.501i$
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: \((2,\ 201,\ (\ :3/2),\ 0.865 + 0.501i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.12536 - 0.302439i\)
\(L(\frac12)\) \(\approx\) \(1.12536 - 0.302439i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-222. - 501. i)T \)
good2 \( 1 + (1.83 + 3.17i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 - 12.9T + 125T^{2} \)
7 \( 1 + (-1.64 + 2.84i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (15.4 - 26.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-34.7 - 60.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-41.6 - 72.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-29.9 - 51.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (40.5 + 70.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-100. + 174. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (24.4 - 42.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-81.7 - 141. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-24.7 + 42.8i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 - 47.5T + 7.95e4T^{2} \)
47 \( 1 + (-29.0 + 50.2i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 538.T + 1.48e5T^{2} \)
59 \( 1 - 85.8T + 2.05e5T^{2} \)
61 \( 1 + (405. + 702. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
71 \( 1 + (-261. + 453. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-444. - 770. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (188. - 325. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (318. + 551. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 873.T + 7.04e5T^{2} \)
97 \( 1 + (-399. - 692. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.80257377082455410483945311499, −10.73301612205128700364847050935, −10.07786944067841996417850106800, −9.438913772706306312146788023213, −8.214610966856789523324219226404, −6.52812912875122334782170376677, −5.71437935933529853059097082497, −4.08072716321895247815597890212, −2.23987505067742325333321639486, −1.30240125668037089391941344704, 0.76119251089062368699439372660, 2.97645515813532904931994819554, 5.54367382518135936018918440400, 5.61972048403107410216424676823, 6.91566493718031957311841335753, 7.956910372423869361465894641357, 9.003956682702162295663413764116, 9.903179387119670080668927897531, 10.86703254450907363370158785569, 12.06254676405933897041275135090

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