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Properties

Label 2-15e2-45.32-c3-0-29
Degree $2$
Conductor $225$
Sign $-0.946 + 0.323i$
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  + (−4.14 − 1.11i)2-s + (−1.80 − 4.87i)3-s + (9.00 + 5.20i)4-s + (2.06 + 22.1i)6-s + (−3.90 + 14.5i)7-s + (−7.28 − 7.28i)8-s + (−20.4 + 17.5i)9-s + (42.5 − 24.5i)11-s + (9.09 − 53.2i)12-s + (−9.04 − 33.7i)13-s + (32.3 − 56.0i)14-s + (−19.5 − 33.7i)16-s + (18.9 − 18.9i)17-s + (104. − 50.0i)18-s + 53.7i·19-s + ⋯
L(s)  = 1  + (−1.46 − 0.392i)2-s + (−0.347 − 0.937i)3-s + (1.12 + 0.650i)4-s + (0.140 + 1.51i)6-s + (−0.211 + 0.787i)7-s + (−0.321 − 0.321i)8-s + (−0.759 + 0.650i)9-s + (1.16 − 0.673i)11-s + (0.218 − 1.28i)12-s + (−0.193 − 0.720i)13-s + (0.618 − 1.07i)14-s + (−0.304 − 0.527i)16-s + (0.269 − 0.269i)17-s + (1.36 − 0.655i)18-s + 0.649i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.946 + 0.323i$
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: \((2,\ 225,\ (\ :3/2),\ -0.946 + 0.323i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0779233 - 0.468217i\)
\(L(\frac12)\) \(\approx\) \(0.0779233 - 0.468217i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.80 + 4.87i)T \)
5 \( 1 \)
good2 \( 1 + (4.14 + 1.11i)T + (6.92 + 4i)T^{2} \)
7 \( 1 + (3.90 - 14.5i)T + (-297. - 171.5i)T^{2} \)
11 \( 1 + (-42.5 + 24.5i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (9.04 + 33.7i)T + (-1.90e3 + 1.09e3i)T^{2} \)
17 \( 1 + (-18.9 + 18.9i)T - 4.91e3iT^{2} \)
19 \( 1 - 53.7iT - 6.85e3T^{2} \)
23 \( 1 + (-170. + 45.7i)T + (1.05e4 - 6.08e3i)T^{2} \)
29 \( 1 + (110. + 190. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (22.1 - 38.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (169. + 169. i)T + 5.06e4iT^{2} \)
41 \( 1 + (-42.1 - 24.3i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (344. + 92.3i)T + (6.88e4 + 3.97e4i)T^{2} \)
47 \( 1 + (512. + 137. i)T + (8.99e4 + 5.19e4i)T^{2} \)
53 \( 1 + (-198. - 198. i)T + 1.48e5iT^{2} \)
59 \( 1 + (64.3 - 111. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (33.4 + 57.9i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-69.7 + 18.6i)T + (2.60e5 - 1.50e5i)T^{2} \)
71 \( 1 + 1.03e3iT - 3.57e5T^{2} \)
73 \( 1 + (339. - 339. i)T - 3.89e5iT^{2} \)
79 \( 1 + (297. - 171. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-225. + 841. i)T + (-4.95e5 - 2.85e5i)T^{2} \)
89 \( 1 + 230.T + 7.04e5T^{2} \)
97 \( 1 + (-148. + 553. i)T + (-7.90e5 - 4.56e5i)T^{2} \)
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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

−11.39267465973852894386892724465, −10.38231568991819898209024153069, −9.204843232034121371699318997780, −8.548218054858856306625455601101, −7.58764621631083043879651658796, −6.53562527334645864823996982241, −5.40992498844278860600434948696, −2.99943171425050005287447137529, −1.63359941276211521734797565942, −0.38278545055731700509493729040, 1.27425738873265536551643537840, 3.66661834897164853828593562386, 4.87757352419664010988922957554, 6.62616650535619179198524564654, 7.13161972804756000275126678070, 8.654528665695623730631180158924, 9.368075270411247446747236821187, 9.989931974260584928911970686775, 10.94441775530700688936897410729, 11.68154874503728651732408631352

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