Properties

Label 2-15e2-25.11-c3-0-7
Degree $2$
Conductor $225$
Sign $-0.753 - 0.657i$
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  + (−3.54 + 2.57i)2-s + (3.45 − 10.6i)4-s + (10.9 − 2.29i)5-s − 21.8·7-s + (4.28 + 13.2i)8-s + (−32.8 + 36.2i)10-s + (17.2 − 12.5i)11-s + (−62.3 − 45.2i)13-s + (77.4 − 56.2i)14-s + (23.1 + 16.8i)16-s + (36.3 + 111. i)17-s + (33.1 + 101. i)19-s + (13.3 − 124. i)20-s + (−28.8 + 88.9i)22-s + (82.9 − 60.2i)23-s + ⋯
L(s)  = 1  + (−1.25 + 0.909i)2-s + (0.431 − 1.32i)4-s + (0.978 − 0.205i)5-s − 1.18·7-s + (0.189 + 0.583i)8-s + (−1.03 + 1.14i)10-s + (0.473 − 0.344i)11-s + (−1.32 − 0.966i)13-s + (1.47 − 1.07i)14-s + (0.361 + 0.262i)16-s + (0.519 + 1.59i)17-s + (0.400 + 1.23i)19-s + (0.149 − 1.38i)20-s + (−0.279 + 0.861i)22-s + (0.751 − 0.546i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.753 - 0.657i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.753 - 0.657i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.223404 + 0.595642i\)
\(L(\frac12)\) \(\approx\) \(0.223404 + 0.595642i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-10.9 + 2.29i)T \)
good2 \( 1 + (3.54 - 2.57i)T + (2.47 - 7.60i)T^{2} \)
7 \( 1 + 21.8T + 343T^{2} \)
11 \( 1 + (-17.2 + 12.5i)T + (411. - 1.26e3i)T^{2} \)
13 \( 1 + (62.3 + 45.2i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (-36.3 - 111. i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (-33.1 - 101. i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 + (-82.9 + 60.2i)T + (3.75e3 - 1.15e4i)T^{2} \)
29 \( 1 + (77.0 - 237. i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (-44.9 - 138. i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (44.0 + 31.9i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (-45.9 - 33.3i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 + 233.T + 7.95e4T^{2} \)
47 \( 1 + (76.4 - 235. i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (46.1 - 141. i)T + (-1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (-235. - 170. i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (181. - 132. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + (-273. - 841. i)T + (-2.43e5 + 1.76e5i)T^{2} \)
71 \( 1 + (-11.8 + 36.5i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (742. - 539. i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-283. + 871. i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (-243. - 750. i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 + (-409. + 297. i)T + (2.17e5 - 6.70e5i)T^{2} \)
97 \( 1 + (-5.01 + 15.4i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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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

−12.52396365745982388896328212934, −10.38897071616989637008272186019, −10.11592925161953749426938390072, −9.188886784932485183390298978206, −8.340709107156077940087632057431, −7.16055524037164445884531303745, −6.24720982202602053861909698148, −5.47078891592071116118228196675, −3.25255902631552625735497470739, −1.27110869093819400898203433751, 0.42768371987317562361916294256, 2.14753933030229695546278119242, 3.04390005088806491904111750645, 5.07517843012129404892448733122, 6.66281498143928071521686224291, 7.43178891905344259604909776370, 9.280200652422792982049234890695, 9.463111628314156631949266354158, 10.00348436297656288503215536437, 11.39010515434002900086608921852

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