↑

Properties

Label 2-42e2-21.17-c3-0-5
Degree $2$
Conductor $1764$
Sign $-0.851 + 0.524i$
Analytic cond. $104.079$
Root an. cond. $10.2019$
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  + (7.66 + 13.2i)5-s + (17.6 + 10.1i)11-s + 30.5i·13-s + (−1.72 + 2.98i)17-s + (−127. + 73.6i)19-s + (−110. + 63.5i)23-s + (−55.0 + 95.3i)25-s − 216. i·29-s + (89.6 + 51.7i)31-s + (−118. − 204. i)37-s − 480.·41-s + 147.·43-s + (−227. − 394. i)47-s + (173. + 100. i)53-s + 311. i·55-s + ⋯
L(s)  = 1  + (0.685 + 1.18i)5-s + (0.482 + 0.278i)11-s + 0.651i·13-s + (−0.0246 + 0.0426i)17-s + (−1.54 + 0.889i)19-s + (−0.998 + 0.576i)23-s + (−0.440 + 0.762i)25-s − 1.38i·29-s + (0.519 + 0.299i)31-s + (−0.525 − 0.910i)37-s − 1.82·41-s + 0.524·43-s + (−0.706 − 1.22i)47-s + (0.450 + 0.260i)53-s + 0.764i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.851 + 0.524i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ -0.851 + 0.524i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5442470488\)
\(L(\frac12)\) \(\approx\) \(0.5442470488\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-7.66 - 13.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-17.6 - 10.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 30.5iT - 2.19e3T^{2} \)
17 \( 1 + (1.72 - 2.98i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (127. - 73.6i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (110. - 63.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 216. iT - 2.43e4T^{2} \)
31 \( 1 + (-89.6 - 51.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (118. + 204. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 480.T + 6.89e4T^{2} \)
43 \( 1 - 147.T + 7.95e4T^{2} \)
47 \( 1 + (227. + 394. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-173. - 100. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-80.1 + 138. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (712. - 411. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-203. + 352. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 513. iT - 3.57e5T^{2} \)
73 \( 1 + (-410. - 236. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-60.6 - 105. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 536.T + 5.71e5T^{2} \)
89 \( 1 + (-115. - 200. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 734. iT - 9.12e5T^{2} \)
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

−9.565601286788820167502661206369, −8.619638403685462753541994496316, −7.78924580556868306398366319407, −6.74275686861224734067165985942, −6.40563686596949169708143261580, −5.57637080143329513677206395739, −4.28073158585101218218040206009, −3.58153066271880721526235316883, −2.30534015670840554098491025868, −1.76674710634667648707852151611, 0.10762165277977245011961541333, 1.20956429166522217343968267048, 2.15156026063210638024930836272, 3.37369936943995010466643385737, 4.57046107269276734003457236379, 5.05472261995061770809475940973, 6.11002898738134846462682895292, 6.64428023339844508541630251604, 7.915193685077430274506514249908, 8.672849333696742328387220538828

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