↑

Properties

Label 2-201-67.37-c3-0-28
Degree $2$
Conductor $201$
Sign $0.722 - 0.691i$
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  + (1.70 + 2.94i)2-s + 3·3-s + (−1.78 + 3.08i)4-s + 10.4·5-s + (5.10 + 8.83i)6-s + (9.10 − 15.7i)7-s + 15.0·8-s + 9·9-s + (17.7 + 30.6i)10-s + (2.36 − 4.09i)11-s + (−5.34 + 9.26i)12-s + (−44.9 − 77.8i)13-s + 61.9·14-s + 31.2·15-s + (39.9 + 69.1i)16-s + (−9.22 − 15.9i)17-s + ⋯
L(s)  = 1  + (0.601 + 1.04i)2-s + 0.577·3-s + (−0.222 + 0.385i)4-s + 0.931·5-s + (0.347 + 0.601i)6-s + (0.491 − 0.851i)7-s + 0.666·8-s + 0.333·9-s + (0.560 + 0.970i)10-s + (0.0647 − 0.112i)11-s + (−0.128 + 0.222i)12-s + (−0.958 − 1.66i)13-s + 1.18·14-s + 0.538·15-s + (0.623 + 1.07i)16-s + (−0.131 − 0.228i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.722 - 0.691i$
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: \((2,\ 201,\ (\ :3/2),\ 0.722 - 0.691i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.36486 + 1.34961i\)
\(L(\frac12)\) \(\approx\) \(3.36486 + 1.34961i\)
\(L(\frac{5}{2})\) 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 \( 1 + (543. - 74.2i)T \)
good2 \( 1 + (-1.70 - 2.94i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 - 10.4T + 125T^{2} \)
7 \( 1 + (-9.10 + 15.7i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-2.36 + 4.09i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (44.9 + 77.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (9.22 + 15.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-7.68 - 13.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-80.8 - 140. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (147. - 255. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (157. - 272. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (100. + 174. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-134. + 232. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + 102.T + 7.95e4T^{2} \)
47 \( 1 + (130. - 225. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 342.T + 1.48e5T^{2} \)
59 \( 1 - 265.T + 2.05e5T^{2} \)
61 \( 1 + (-395. - 684. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
71 \( 1 + (-212. + 367. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-249. - 432. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-603. + 1.04e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (632. + 1.09e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 818.T + 7.04e5T^{2} \)
97 \( 1 + (134. + 233. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−12.71022874282666083811569980823, −10.84585727862214042509548958234, −10.19091805203812517114641237534, −8.990981113239138946848131562563, −7.56418778007559839474588199376, −7.20867422349730350222421794634, −5.63351518761328309079477874489, −5.01098581720662938991698527482, −3.39927564551611336415142590661, −1.54978889308732214865881488275, 1.98838279045922011094916336811, 2.35968107756120110690164778096, 4.09361792719878828491851382001, 5.11676608197661188304392880772, 6.60183053637595810868876955367, 7.964137720737612025759409576786, 9.296983548368535999836465078393, 9.825043059012514232369661087268, 11.21584211435572247303952977630, 11.86058480944933471641975807370

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