Properties

Label 2-201-67.38-c2-0-12
Degree $2$
Conductor $201$
Sign $0.916 + 0.400i$
Analytic cond. $5.47685$
Root an. cond. $2.34026$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0720 + 0.0416i)2-s + 1.73i·3-s + (−1.99 − 3.45i)4-s + 1.32i·5-s + (−0.0720 + 0.124i)6-s + (1.88 − 1.08i)7-s − 0.665i·8-s − 2.99·9-s + (−0.0549 + 0.0951i)10-s + (18.4 − 10.6i)11-s + (5.98 − 3.45i)12-s + (6.76 + 3.90i)13-s + 0.180·14-s − 2.28·15-s + (−7.95 + 13.7i)16-s + (13.7 − 23.8i)17-s + ⋯
L(s)  = 1  + (0.0360 + 0.0208i)2-s + 0.577i·3-s + (−0.499 − 0.864i)4-s + 0.264i·5-s + (−0.0120 + 0.0208i)6-s + (0.268 − 0.155i)7-s − 0.0831i·8-s − 0.333·9-s + (−0.00549 + 0.00951i)10-s + (1.67 − 0.967i)11-s + (0.499 − 0.288i)12-s + (0.520 + 0.300i)13-s + 0.0129·14-s − 0.152·15-s + (−0.497 + 0.861i)16-s + (0.811 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ 0.916 + 0.400i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.48876 - 0.311106i\)
\(L(\frac12)\) \(\approx\) \(1.48876 - 0.311106i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
67 \( 1 + (-27.8 - 60.9i)T \)
good2 \( 1 + (-0.0720 - 0.0416i)T + (2 + 3.46i)T^{2} \)
5 \( 1 - 1.32iT - 25T^{2} \)
7 \( 1 + (-1.88 + 1.08i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-18.4 + 10.6i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-6.76 - 3.90i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-13.7 + 23.8i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.95 + 6.85i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (11.7 - 20.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (10.4 + 18.1i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-34.8 + 20.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (26.1 - 45.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (8.87 - 5.12i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 3.99iT - 1.84e3T^{2} \)
47 \( 1 + (27.2 + 47.1i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 18.5iT - 2.80e3T^{2} \)
59 \( 1 - 18.6T + 3.48e3T^{2} \)
61 \( 1 + (41.8 + 24.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
71 \( 1 + (43.8 + 75.8i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (22.6 - 39.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (129. - 74.7i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (72.7 - 125. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 139.T + 7.92e3T^{2} \)
97 \( 1 + (-25.9 - 14.9i)T + (4.70e3 + 8.14e3i)T^{2} \)
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

−11.69275136888865953990987750275, −11.31647297505676672653114320797, −10.01959535221251404148243687360, −9.358356381901825752915220614528, −8.440139800784296371557129057266, −6.77297922868150289079082267670, −5.77820779620609125654077512228, −4.61143802558028647582937677230, −3.42622132571257278737818265462, −1.07855043408494212500386945012, 1.51848726502060492819346450423, 3.47962762027463722449255886453, 4.58422368222963278460606323746, 6.15501307810876356165850089542, 7.27222956195205603237193130927, 8.370013717421350251896120710827, 8.977267838196544925201205655470, 10.29997566375776431323180094873, 11.73692248072803054807230638083, 12.40335205151601859135689683596

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