↑

Properties

Label 2-201-201.8-c3-0-45
Degree $2$
Conductor $201$
Sign $0.951 + 0.306i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (3.92 − 3.40i)3-s + (7.67 + 2.25i)4-s + (35.1 + 5.05i)7-s + (3.84 − 26.7i)9-s + (37.8 − 17.2i)12-s + (−71.0 + 32.4i)13-s + (53.8 + 34.6i)16-s + (11.5 + 80.3i)19-s + (155. − 99.8i)21-s + (−51.9 − 113. i)25-s + (−75.8 − 118. i)27-s + (258. + 118. i)28-s + (−270. − 123. i)31-s + (89.7 − 196. i)36-s + 351.·37-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)3-s + (0.959 + 0.281i)4-s + (1.89 + 0.273i)7-s + (0.142 − 0.989i)9-s + (0.909 − 0.415i)12-s + (−1.51 + 0.692i)13-s + (0.841 + 0.540i)16-s + (0.139 + 0.969i)19-s + (1.61 − 1.03i)21-s + (−0.415 − 0.909i)25-s + (−0.540 − 0.841i)27-s + (1.74 + 0.797i)28-s + (−1.56 − 0.716i)31-s + (0.415 − 0.909i)36-s + 1.56·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.951 + 0.306i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ 0.951 + 0.306i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.09741 - 0.486777i\)
\(L(\frac12)\) \(\approx\) \(3.09741 - 0.486777i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.92 + 3.40i)T \)
67 \( 1 + (114. - 536. i)T \)
good2 \( 1 + (-7.67 - 2.25i)T^{2} \)
5 \( 1 + (51.9 + 113. i)T^{2} \)
7 \( 1 + (-35.1 - 5.05i)T + (329. + 96.6i)T^{2} \)
11 \( 1 + (552. + 1.21e3i)T^{2} \)
13 \( 1 + (71.0 - 32.4i)T + (1.43e3 - 1.66e3i)T^{2} \)
17 \( 1 + (-4.13e3 + 2.65e3i)T^{2} \)
19 \( 1 + (-11.5 - 80.3i)T + (-6.58e3 + 1.93e3i)T^{2} \)
23 \( 1 + (1.73e3 - 1.20e4i)T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (270. + 123. i)T + (1.95e4 + 2.25e4i)T^{2} \)
37 \( 1 - 351.T + 5.06e4T^{2} \)
41 \( 1 + (5.79e4 - 3.72e4i)T^{2} \)
43 \( 1 + (107. + 366. i)T + (-6.68e4 + 4.29e4i)T^{2} \)
47 \( 1 + (1.47e4 - 1.02e5i)T^{2} \)
53 \( 1 + (1.25e5 + 8.04e4i)T^{2} \)
59 \( 1 + (1.34e5 + 1.55e5i)T^{2} \)
61 \( 1 + (256. + 399. i)T + (-9.42e4 + 2.06e5i)T^{2} \)
71 \( 1 + (-3.01e5 - 1.93e5i)T^{2} \)
73 \( 1 + (1.01e3 - 650. i)T + (1.61e5 - 3.53e5i)T^{2} \)
79 \( 1 + (1.14e3 - 522. i)T + (3.22e5 - 3.72e5i)T^{2} \)
83 \( 1 + (-2.37e5 - 5.20e5i)T^{2} \)
89 \( 1 + (1.00e5 + 6.97e5i)T^{2} \)
97 \( 1 - 106. iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.89937049135786886558021234464, −11.34586002357458513776315253768, −9.968753010703190424625992176316, −8.622944473570627366154357098239, −7.74124165651500603229616495098, −7.23580465484811899988623396754, −5.75154744077751712678828394731, −4.21326452286733522600341823087, −2.41824532651759082946038680030, −1.70237081698586924321910898859, 1.68866184438205825505810645228, 2.84872492312082873676205305338, 4.61288561235053837724838966349, 5.39309455261486496060917106663, 7.44563879882219684295804605952, 7.72280395770051373075203380023, 9.097182559926102044633973629115, 10.23327321507199671505339131707, 11.03001341482568068436952401556, 11.69103120419971358159521939711

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