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Properties

Label 2-15e2-45.4-c3-0-50
Degree $2$
Conductor $225$
Sign $-0.836 + 0.547i$
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  + (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

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.836 + 0.547i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.836 + 0.547i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.33978 - 4.49668i\)
\(L(\frac12)\) \(\approx\) \(1.33978 - 4.49668i\)
\(L(\frac{5}{2})\) 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 \( 1 + (763. - 440. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.56782816149391953094197013422, −11.07804455666726033791755435141, −9.871275983569538329848912571804, −8.275866369120522468091087948825, −6.81297977677022330245668479967, −6.16978980580587250547125560452, −4.77077053909056228154250010788, −3.82254213423838761759639461303, −2.20861894458825557214975791032, −1.36391725151365134587599556523, 2.85824278398779058404534158212, 3.89347433220955960054633929511, 4.88141120198828302897984366287, 5.75977127372274041766048740454, 6.65616922590436029217586547575, 8.252096733075097880005069968742, 8.778890621751168732086529269061, 10.66903925095188480945276151239, 11.46519753415065972151699971757, 12.25295532819751289054455975331

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