Properties

Label 2-3e2-1.1-c73-0-5
Degree $2$
Conductor $9$
Sign $1$
Analytic cond. $303.735$
Root an. cond. $17.4280$
Motivic weight $73$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.59e11·2-s + 1.59e22·4-s − 2.70e25·5-s − 5.83e30·7-s + 1.04e33·8-s − 4.31e36·10-s − 1.33e38·11-s − 6.53e40·13-s − 9.30e41·14-s + 1.51e43·16-s − 1.34e45·17-s + 4.71e46·19-s − 4.32e47·20-s − 2.12e49·22-s + 7.28e48·23-s − 3.25e50·25-s − 1.04e52·26-s − 9.32e52·28-s + 1.30e53·29-s − 6.62e53·31-s − 7.42e54·32-s − 2.14e56·34-s + 1.58e56·35-s + 1.59e57·37-s + 7.51e57·38-s − 2.82e58·40-s + 1.09e59·41-s + ⋯
L(s)  = 1  + 1.64·2-s + 1.69·4-s − 0.832·5-s − 0.832·7-s + 1.13·8-s − 1.36·10-s − 1.29·11-s − 1.43·13-s − 1.36·14-s + 0.169·16-s − 1.65·17-s + 0.997·19-s − 1.40·20-s − 2.13·22-s + 0.144·23-s − 0.307·25-s − 2.35·26-s − 1.40·28-s + 0.548·29-s − 0.243·31-s − 0.856·32-s − 2.71·34-s + 0.692·35-s + 0.916·37-s + 1.63·38-s − 0.944·40-s + 1.48·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(74-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+73/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(303.735\)
Root analytic conductor: \(17.4280\)
Motivic weight: \(73\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :73/2),\ 1)\)

Particular Values

\(L(37)\) \(\approx\) \(1.919496499\)
\(L(\frac12)\) \(\approx\) \(1.919496499\)
\(L(\frac{75}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 1.59e11T + 9.44e21T^{2} \)
5 \( 1 + 2.70e25T + 1.05e51T^{2} \)
7 \( 1 + 5.83e30T + 4.92e61T^{2} \)
11 \( 1 + 1.33e38T + 1.05e76T^{2} \)
13 \( 1 + 6.53e40T + 2.07e81T^{2} \)
17 \( 1 + 1.34e45T + 6.64e89T^{2} \)
19 \( 1 - 4.71e46T + 2.23e93T^{2} \)
23 \( 1 - 7.28e48T + 2.54e99T^{2} \)
29 \( 1 - 1.30e53T + 5.68e106T^{2} \)
31 \( 1 + 6.62e53T + 7.40e108T^{2} \)
37 \( 1 - 1.59e57T + 3.01e114T^{2} \)
41 \( 1 - 1.09e59T + 5.41e117T^{2} \)
43 \( 1 - 6.52e59T + 1.75e119T^{2} \)
47 \( 1 + 6.06e60T + 1.15e122T^{2} \)
53 \( 1 - 1.02e63T + 7.44e125T^{2} \)
59 \( 1 - 2.47e64T + 1.87e129T^{2} \)
61 \( 1 + 8.75e64T + 2.13e130T^{2} \)
67 \( 1 + 2.58e66T + 2.01e133T^{2} \)
71 \( 1 - 4.91e67T + 1.38e135T^{2} \)
73 \( 1 + 1.02e68T + 1.05e136T^{2} \)
79 \( 1 + 2.22e69T + 3.36e138T^{2} \)
83 \( 1 - 1.44e70T + 1.23e140T^{2} \)
89 \( 1 + 3.70e70T + 2.02e142T^{2} \)
97 \( 1 - 3.38e71T + 1.08e145T^{2} \)
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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

−10.83374323203884171842763003621, −9.438207678232621215783012924067, −7.72108339797097042515346697283, −6.93004335246011958480654269154, −5.74753524637386963497564344269, −4.77179411794378891864422554027, −4.03928393009269085365317674770, −2.83575456804906773712840148775, −2.42525346487783307427463597595, −0.37736176993508001987274472342, 0.37736176993508001987274472342, 2.42525346487783307427463597595, 2.83575456804906773712840148775, 4.03928393009269085365317674770, 4.77179411794378891864422554027, 5.74753524637386963497564344269, 6.93004335246011958480654269154, 7.72108339797097042515346697283, 9.438207678232621215783012924067, 10.83374323203884171842763003621

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