Properties

Label 2-15e2-9.4-c3-0-22
Degree $2$
Conductor $225$
Sign $-0.821 - 0.570i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.51 + 4.35i)2-s + (4.79 − 1.99i)3-s + (−8.61 + 14.9i)4-s + (20.7 + 15.8i)6-s + (2.69 + 4.66i)7-s − 46.4·8-s + (19.0 − 19.1i)9-s + (19.5 + 33.8i)11-s + (−11.5 + 88.8i)12-s + (−43.3 + 75.0i)13-s + (−13.5 + 23.4i)14-s + (−47.6 − 82.4i)16-s + 15.4·17-s + (131. + 34.6i)18-s − 26.8·19-s + ⋯
L(s)  = 1  + (0.888 + 1.53i)2-s + (0.923 − 0.384i)3-s + (−1.07 + 1.86i)4-s + (1.41 + 1.07i)6-s + (0.145 + 0.251i)7-s − 2.05·8-s + (0.704 − 0.709i)9-s + (0.535 + 0.928i)11-s + (−0.277 + 2.13i)12-s + (−0.924 + 1.60i)13-s + (−0.258 + 0.446i)14-s + (−0.744 − 1.28i)16-s + 0.219·17-s + (1.71 + 0.453i)18-s − 0.324·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.821 - 0.570i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.821 - 0.570i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.06226 + 3.38875i\)
\(L(\frac12)\) \(\approx\) \(1.06226 + 3.38875i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.79 + 1.99i)T \)
5 \( 1 \)
good2 \( 1 + (-2.51 - 4.35i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-2.69 - 4.66i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-19.5 - 33.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (43.3 - 75.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 15.4T + 4.91e3T^{2} \)
19 \( 1 + 26.8T + 6.85e3T^{2} \)
23 \( 1 + (-55.5 + 96.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (24.6 + 42.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-89.9 + 155. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 293.T + 5.06e4T^{2} \)
41 \( 1 + (13.8 - 23.9i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (30.2 + 52.3i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (48.1 + 83.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 251.T + 1.48e5T^{2} \)
59 \( 1 + (-38.4 + 66.5i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (245. + 424. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-119. + 206. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 640.T + 3.57e5T^{2} \)
73 \( 1 - 769.T + 3.89e5T^{2} \)
79 \( 1 + (331. + 573. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-651. - 1.12e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 995.T + 7.04e5T^{2} \)
97 \( 1 + (-407. - 705. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.53671679174302983807868832912, −11.82686271716126590774702778443, −9.699356424341200121181020986293, −8.912186250491321554619544355330, −7.87718321179021042105021916512, −7.01496166952660334154056919959, −6.38517110837820761476782744877, −4.74591375358799176192040287307, −4.02379117558964563781633386064, −2.27179990763829296685727652133, 1.09778990511150579956733406734, 2.71545742517123333872573657817, 3.42979051421083818282430930650, 4.58820055170012245853434631034, 5.63736885795460478260147930956, 7.59038228271174231086301861662, 8.806740280073500271622841032381, 9.848811756393386309585171004518, 10.51728164126824982430962057911, 11.37192322245696830144541087424

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