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Properties

Label 2-201-67.37-c3-0-22
Degree $2$
Conductor $201$
Sign $0.983 - 0.182i$
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  + (0.660 + 1.14i)2-s + 3·3-s + (3.12 − 5.41i)4-s + 15.1·5-s + (1.98 + 3.43i)6-s + (−6.71 + 11.6i)7-s + 18.8·8-s + 9·9-s + (9.97 + 17.2i)10-s + (−0.0183 + 0.0318i)11-s + (9.38 − 16.2i)12-s + (−3.98 − 6.89i)13-s − 17.7·14-s + 45.3·15-s + (−12.5 − 21.7i)16-s + (50.7 + 87.8i)17-s + ⋯
L(s)  = 1  + (0.233 + 0.404i)2-s + 0.577·3-s + (0.390 − 0.676i)4-s + 1.35·5-s + (0.134 + 0.233i)6-s + (−0.362 + 0.627i)7-s + 0.832·8-s + 0.333·9-s + (0.315 + 0.546i)10-s + (−0.000504 + 0.000873i)11-s + (0.225 − 0.390i)12-s + (−0.0849 − 0.147i)13-s − 0.338·14-s + 0.779·15-s + (−0.196 − 0.340i)16-s + (0.723 + 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.983 - 0.182i$
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: \((2,\ 201,\ (\ :3/2),\ 0.983 - 0.182i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.22046 + 0.295574i\)
\(L(\frac12)\) \(\approx\) \(3.22046 + 0.295574i\)
\(L(\frac{5}{2})\) 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 \( 1 + (495. + 235. i)T \)
good2 \( 1 + (-0.660 - 1.14i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 - 15.1T + 125T^{2} \)
7 \( 1 + (6.71 - 11.6i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (0.0183 - 0.0318i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (3.98 + 6.89i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-50.7 - 87.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (38.6 + 66.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (61.5 + 106. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-82.5 + 143. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (81.4 - 141. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-124. - 215. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (209. - 363. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 - 266.T + 7.95e4T^{2} \)
47 \( 1 + (-124. + 215. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 612.T + 1.48e5T^{2} \)
59 \( 1 + 617.T + 2.05e5T^{2} \)
61 \( 1 + (-25.1 - 43.5i)T + (-1.13e5 + 1.96e5i)T^{2} \)
71 \( 1 + (-229. + 397. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (246. + 427. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-426. + 738. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-0.417 - 0.723i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.18e3T + 7.04e5T^{2} \)
97 \( 1 + (-224. - 389. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.28731038427056756570925305801, −10.70035268795014531006617264832, −10.02222435330734706182841655619, −9.181926874568237786924252122924, −7.988448421494920134026682483691, −6.42715354153523539366736357399, −6.02177747580346448497553477835, −4.74110498625015972315942298619, −2.72040668546836398607140578513, −1.61996068993736215581283326225, 1.67336790662425321373353755647, 2.84639173178645878767168587944, 4.02249446498374776171446021530, 5.62941138980278084912812949079, 6.95933838989954737997531115703, 7.81024081765849621161103140958, 9.229882435856647687193736424799, 9.979233486602486758694513369153, 10.90566193169538874299755608257, 12.19767754879373534367471115794

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