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Properties

Label 2-201-67.29-c3-0-1
Degree $2$
Conductor $201$
Sign $-0.773 + 0.634i$
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  + (−1.56 + 2.70i)2-s − 3·3-s + (−0.869 − 1.50i)4-s + 10.0·5-s + (4.68 − 8.10i)6-s + (−2.32 − 4.02i)7-s − 19.5·8-s + 9·9-s + (−15.7 + 27.2i)10-s + (5.45 + 9.45i)11-s + (2.60 + 4.51i)12-s + (−20.3 + 35.3i)13-s + 14.5·14-s − 30.2·15-s + (37.4 − 64.8i)16-s + (−34.9 + 60.4i)17-s + ⋯
L(s)  = 1  + (−0.551 + 0.955i)2-s − 0.577·3-s + (−0.108 − 0.188i)4-s + 0.902·5-s + (0.318 − 0.551i)6-s + (−0.125 − 0.217i)7-s − 0.863·8-s + 0.333·9-s + (−0.497 + 0.862i)10-s + (0.149 + 0.259i)11-s + (0.0627 + 0.108i)12-s + (−0.434 + 0.753i)13-s + 0.277·14-s − 0.521·15-s + (0.585 − 1.01i)16-s + (−0.497 + 0.862i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.773 + 0.634i$
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: \((2,\ 201,\ (\ :3/2),\ -0.773 + 0.634i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.158148 - 0.441988i\)
\(L(\frac12)\) \(\approx\) \(0.158148 - 0.441988i\)
\(L(\frac{5}{2})\) 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 \( 1 + (139. - 530. i)T \)
good2 \( 1 + (1.56 - 2.70i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 10.0T + 125T^{2} \)
7 \( 1 + (2.32 + 4.02i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-5.45 - 9.45i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (20.3 - 35.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (34.9 - 60.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (62.2 - 107. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-37.3 + 64.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (141. + 244. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (85.2 + 147. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (216. - 375. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (58.4 + 101. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 - 366.T + 7.95e4T^{2} \)
47 \( 1 + (65.8 + 113. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 229.T + 1.48e5T^{2} \)
59 \( 1 + 349.T + 2.05e5T^{2} \)
61 \( 1 + (396. - 687. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (131. + 227. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (329. - 570. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-248. - 430. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-102. + 177. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 347.T + 7.04e5T^{2} \)
97 \( 1 + (-174. + 302. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.56154087725116910288227551035, −11.65956646089342077627454503844, −10.33656565032798411276148447873, −9.579739627114281261896055588025, −8.528448857716074088514346808771, −7.37577958554556489003775124331, −6.36179204379234229912045283525, −5.79212587908118657528584422224, −4.14692094147386199834544806007, −2.00871012481241560502944284964, 0.24309883458928755744771578828, 1.80442656083185743239933638499, 3.05428420176148207871895452433, 5.09185945165171373638256884672, 6.02274268340214608725703840503, 7.19526929281368965832830685241, 9.080343007740253870266701401274, 9.385846808487026404630225406659, 10.77757559064224210522337130216, 10.95012037005521697528779138448

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