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Properties

Label 2-15e2-45.32-c3-0-4
Degree $2$
Conductor $225$
Sign $-0.508 - 0.861i$
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  + (1.60 + 0.429i)2-s + (1.67 − 4.91i)3-s + (−4.54 − 2.62i)4-s + (4.79 − 7.15i)6-s + (−8.15 + 30.4i)7-s + (−15.5 − 15.5i)8-s + (−21.3 − 16.5i)9-s + (−31.0 + 17.9i)11-s + (−20.5 + 17.9i)12-s + (14.3 + 53.6i)13-s + (−26.1 + 45.2i)14-s + (2.79 + 4.84i)16-s + (29.9 − 29.9i)17-s + (−27.1 − 35.6i)18-s − 17.7i·19-s + ⋯
L(s)  = 1  + (0.566 + 0.151i)2-s + (0.323 − 0.946i)3-s + (−0.568 − 0.328i)4-s + (0.326 − 0.486i)6-s + (−0.440 + 1.64i)7-s + (−0.686 − 0.686i)8-s + (−0.791 − 0.611i)9-s + (−0.850 + 0.490i)11-s + (−0.494 + 0.431i)12-s + (0.306 + 1.14i)13-s + (−0.498 + 0.863i)14-s + (0.0436 + 0.0756i)16-s + (0.427 − 0.427i)17-s + (−0.355 − 0.466i)18-s − 0.213i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 - 0.861i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.508 - 0.861i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.508 - 0.861i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.298350 + 0.522718i\)
\(L(\frac12)\) \(\approx\) \(0.298350 + 0.522718i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 + 4.91i)T \)
5 \( 1 \)
good2 \( 1 + (-1.60 - 0.429i)T + (6.92 + 4i)T^{2} \)
7 \( 1 + (8.15 - 30.4i)T + (-297. - 171.5i)T^{2} \)
11 \( 1 + (31.0 - 17.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-14.3 - 53.6i)T + (-1.90e3 + 1.09e3i)T^{2} \)
17 \( 1 + (-29.9 + 29.9i)T - 4.91e3iT^{2} \)
19 \( 1 + 17.7iT - 6.85e3T^{2} \)
23 \( 1 + (101. - 27.3i)T + (1.05e4 - 6.08e3i)T^{2} \)
29 \( 1 + (-26.5 - 46.0i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (19.1 - 33.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-75.1 - 75.1i)T + 5.06e4iT^{2} \)
41 \( 1 + (126. + 73.2i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (526. + 140. i)T + (6.88e4 + 3.97e4i)T^{2} \)
47 \( 1 + (161. + 43.2i)T + (8.99e4 + 5.19e4i)T^{2} \)
53 \( 1 + (170. + 170. i)T + 1.48e5iT^{2} \)
59 \( 1 + (28.0 - 48.5i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-187. - 324. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (110. - 29.6i)T + (2.60e5 - 1.50e5i)T^{2} \)
71 \( 1 + 921. iT - 3.57e5T^{2} \)
73 \( 1 + (373. - 373. i)T - 3.89e5iT^{2} \)
79 \( 1 + (-492. + 284. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (117. - 439. i)T + (-4.95e5 - 2.85e5i)T^{2} \)
89 \( 1 - 148.T + 7.04e5T^{2} \)
97 \( 1 + (83.7 - 312. i)T + (-7.90e5 - 4.56e5i)T^{2} \)
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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

−12.25420459821495141911979239221, −11.72311478300467512095838360716, −9.887966263829714975717457327709, −9.072434683462585278944081207590, −8.277017495534649389575445760538, −6.81199329614273249890066178546, −5.92838971690030148178494451939, −5.00601795311657444932499575379, −3.26753788608700914672601924819, −1.96837181133712698141004053218, 0.19130730043330431669138197591, 3.11356647717320387082788192862, 3.78163226612061301576574893609, 4.82617715486282689462362751702, 5.95148578924161326599224104269, 7.81631747216447398997393098116, 8.349278187642499336160125396090, 9.865681014195943761257473681232, 10.34256664509334864646113852369, 11.32304828338840043648771218478

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