Properties

Label 2-15e2-5.4-c9-0-5
Degree $2$
Conductor $225$
Sign $-0.894 + 0.447i$
Analytic cond. $115.883$
Root an. cond. $10.7648$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 20.2i·2-s + 103.·4-s − 4.01e3i·7-s + 1.24e4i·8-s + 4.21e4·11-s − 1.23e5i·13-s + 8.10e4·14-s − 1.98e5·16-s − 3.19e5i·17-s − 1.08e6·19-s + 8.50e5i·22-s + 1.50e6i·23-s + 2.49e6·26-s − 4.16e5i·28-s − 2.62e6·29-s + ⋯
L(s)  = 1  + 0.892i·2-s + 0.202·4-s − 0.631i·7-s + 1.07i·8-s + 0.867·11-s − 1.20i·13-s + 0.563·14-s − 0.755·16-s − 0.929i·17-s − 1.91·19-s + 0.774i·22-s + 1.12i·23-s + 1.07·26-s − 0.128i·28-s − 0.688·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(115.883\)
Root analytic conductor: \(10.7648\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :9/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.5722703796\)
\(L(\frac12)\) \(\approx\) \(0.5722703796\)
\(L(\frac{11}{2})\) 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 \( 1 \)
good2 \( 1 - 20.2iT - 512T^{2} \)
7 \( 1 + 4.01e3iT - 4.03e7T^{2} \)
11 \( 1 - 4.21e4T + 2.35e9T^{2} \)
13 \( 1 + 1.23e5iT - 1.06e10T^{2} \)
17 \( 1 + 3.19e5iT - 1.18e11T^{2} \)
19 \( 1 + 1.08e6T + 3.22e11T^{2} \)
23 \( 1 - 1.50e6iT - 1.80e12T^{2} \)
29 \( 1 + 2.62e6T + 1.45e13T^{2} \)
31 \( 1 - 3.27e6T + 2.64e13T^{2} \)
37 \( 1 + 2.51e6iT - 1.29e14T^{2} \)
41 \( 1 + 2.95e7T + 3.27e14T^{2} \)
43 \( 1 - 1.42e7iT - 5.02e14T^{2} \)
47 \( 1 + 1.35e6iT - 1.11e15T^{2} \)
53 \( 1 - 9.73e7iT - 3.29e15T^{2} \)
59 \( 1 + 7.48e6T + 8.66e15T^{2} \)
61 \( 1 + 9.11e7T + 1.16e16T^{2} \)
67 \( 1 - 2.94e8iT - 2.72e16T^{2} \)
71 \( 1 + 1.56e8T + 4.58e16T^{2} \)
73 \( 1 - 2.82e8iT - 5.88e16T^{2} \)
79 \( 1 - 5.55e8T + 1.19e17T^{2} \)
83 \( 1 - 6.48e6iT - 1.86e17T^{2} \)
89 \( 1 + 5.99e8T + 3.50e17T^{2} \)
97 \( 1 - 9.25e8iT - 7.60e17T^{2} \)
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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

−11.11023173377356189243038539694, −10.25765264461688045998648429972, −8.984677206705379541848375034001, −7.990107378930929177350176640562, −7.15465390636717905947840028875, −6.30750863528121525709894630709, −5.30858996331472713980750482857, −4.06054840265851292290251071992, −2.70665376275146823749822563895, −1.32481699074159740921739054709, 0.10635274859940219100692741282, 1.66626644936066422820405993207, 2.20513588346844288520838154655, 3.62321642007680720238077254826, 4.49059109712202586652093410781, 6.28442548294851923057083311972, 6.71849839745672453779846016418, 8.410494497914595323859870286657, 9.178473287338272218384108285380, 10.28901928991713050675256779388

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