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Properties

Label 2-15e2-5.4-c11-0-40
Degree $2$
Conductor $225$
Sign $-0.447 - 0.894i$
Analytic cond. $172.877$
Root an. cond. $13.1482$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 63.7i·2-s − 2.01e3·4-s − 4.19e4i·7-s + 2.20e3i·8-s + 9.57e5·11-s + 1.39e6i·13-s + 2.67e6·14-s − 4.26e6·16-s + 3.76e6i·17-s − 9.41e6·19-s + 6.10e7i·22-s − 3.02e7i·23-s − 8.86e7·26-s + 8.44e7i·28-s + 1.03e8·29-s + ⋯
L(s)  = 1  + 1.40i·2-s − 0.983·4-s − 0.942i·7-s + 0.0237i·8-s + 1.79·11-s + 1.03i·13-s + 1.32·14-s − 1.01·16-s + 0.643i·17-s − 0.871·19-s + 2.52i·22-s − 0.981i·23-s − 1.46·26-s + 0.926i·28-s + 0.937·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(172.877\)
Root analytic conductor: \(13.1482\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :11/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.515069304\)
\(L(\frac12)\) \(\approx\) \(2.515069304\)
\(L(\frac{13}{2})\) 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 \( 1 - 63.7iT - 2.04e3T^{2} \)
7 \( 1 + 4.19e4iT - 1.97e9T^{2} \)
11 \( 1 - 9.57e5T + 2.85e11T^{2} \)
13 \( 1 - 1.39e6iT - 1.79e12T^{2} \)
17 \( 1 - 3.76e6iT - 3.42e13T^{2} \)
19 \( 1 + 9.41e6T + 1.16e14T^{2} \)
23 \( 1 + 3.02e7iT - 9.52e14T^{2} \)
29 \( 1 - 1.03e8T + 1.22e16T^{2} \)
31 \( 1 + 5.48e7T + 2.54e16T^{2} \)
37 \( 1 + 4.78e8iT - 1.77e17T^{2} \)
41 \( 1 - 9.29e8T + 5.50e17T^{2} \)
43 \( 1 + 2.68e7iT - 9.29e17T^{2} \)
47 \( 1 + 1.20e9iT - 2.47e18T^{2} \)
53 \( 1 - 4.02e9iT - 9.26e18T^{2} \)
59 \( 1 - 7.97e9T + 3.01e19T^{2} \)
61 \( 1 + 2.07e9T + 4.35e19T^{2} \)
67 \( 1 + 5.61e9iT - 1.22e20T^{2} \)
71 \( 1 + 1.51e10T + 2.31e20T^{2} \)
73 \( 1 + 6.64e9iT - 3.13e20T^{2} \)
79 \( 1 - 1.57e10T + 7.47e20T^{2} \)
83 \( 1 - 2.04e10iT - 1.28e21T^{2} \)
89 \( 1 + 4.21e10T + 2.77e21T^{2} \)
97 \( 1 + 1.10e11iT - 7.15e21T^{2} \)
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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

−10.52029481940747433674946162381, −9.175387790907301224269875513237, −8.576938390855742822980815280980, −7.34665574840285871015225811856, −6.65335780868797008196225426179, −6.05095093366536032080627795486, −4.44849592249475814622229958704, −3.99401561358937485172684093768, −2.03187065974828544409462811385, −0.77210174862532291434391745737, 0.65435434685962109062112904877, 1.54349951267763020048305665459, 2.59046360338140322427379292204, 3.47083626543787426310353834279, 4.50998906466315721764440868686, 5.84705348151016428413046841557, 6.91225852503154036294341562168, 8.460758857658752974402262057819, 9.281636037143643047666655785169, 10.00672170972875680529015843504

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