Properties

Label 2-15e2-45.34-c3-0-3
Degree $2$
Conductor $225$
Sign $0.940 - 0.341i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
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  + (−4.60 − 2.65i)2-s + (−0.153 − 5.19i)3-s + (10.1 + 17.5i)4-s + (−13.0 + 24.3i)6-s + (−11.6 − 6.71i)7-s − 65.1i·8-s + (−26.9 + 1.59i)9-s + (23.4 − 40.6i)11-s + (89.5 − 55.2i)12-s + (−31.2 + 18.0i)13-s + (35.7 + 61.8i)14-s + (−92.1 + 159. i)16-s + 54.6i·17-s + (128. + 64.3i)18-s − 111.·19-s + ⋯
L(s)  = 1  + (−1.62 − 0.939i)2-s + (−0.0295 − 0.999i)3-s + (1.26 + 2.19i)4-s + (−0.891 + 1.65i)6-s + (−0.628 − 0.362i)7-s − 2.87i·8-s + (−0.998 + 0.0589i)9-s + (0.643 − 1.11i)11-s + (2.15 − 1.33i)12-s + (−0.667 + 0.385i)13-s + (0.681 + 1.18i)14-s + (−1.43 + 2.49i)16-s + 0.779i·17-s + (1.68 + 0.841i)18-s − 1.34·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.341i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.940 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.940 - 0.341i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.940 - 0.341i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.198531 + 0.0349125i\)
\(L(\frac12)\) \(\approx\) \(0.198531 + 0.0349125i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.153 + 5.19i)T \)
5 \( 1 \)
good2 \( 1 + (4.60 + 2.65i)T + (4 + 6.92i)T^{2} \)
7 \( 1 + (11.6 + 6.71i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-23.4 + 40.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (31.2 - 18.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 54.6iT - 4.91e3T^{2} \)
19 \( 1 + 111.T + 6.85e3T^{2} \)
23 \( 1 + (31.1 - 17.9i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-29.0 + 50.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-147. - 256. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 53.0iT - 5.06e4T^{2} \)
41 \( 1 + (64.1 + 111. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-142. - 82.0i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-76.0 - 43.9i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 479. iT - 1.48e5T^{2} \)
59 \( 1 + (-317. - 550. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (24.0 - 41.5i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-25.0 + 14.4i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 576.T + 3.57e5T^{2} \)
73 \( 1 - 835. iT - 3.89e5T^{2} \)
79 \( 1 + (101. - 176. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (402. + 232. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 993.T + 7.04e5T^{2} \)
97 \( 1 + (-763. - 440. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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

−11.70683315625326575655404229676, −10.82918378581940109457379431296, −9.954918685079034073483289177045, −8.748841073354761908828667254479, −8.289274248385394761133284846654, −7.03012537622960679358305089007, −6.32970591169705392339236469032, −3.63761182380776245324214243187, −2.38889983326841613748506958724, −1.08405749984536695037743468904, 0.16341473418201069118843058613, 2.39560941459091415239874861166, 4.57157783590434326619316593445, 5.89126829566606341019705119445, 6.79538453171414724855556563108, 7.945607232242528150335951271516, 9.006659528348536243876047923479, 9.653019954019104699756006833125, 10.19900940502950386228254478852, 11.25782540163352762405254556050

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