Properties

Label 2-1-1.1-c81-0-3
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $41.5501$
Root an. cond. $6.44593$
Motivic weight $81$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.81e11·2-s − 1.66e18·3-s − 2.27e24·4-s + 2.26e28·5-s − 6.33e29·6-s − 1.01e33·7-s − 1.78e36·8-s − 4.40e38·9-s + 8.62e39·10-s + 2.49e42·11-s + 3.77e42·12-s − 4.84e44·13-s − 3.88e44·14-s − 3.76e46·15-s + 4.81e48·16-s − 5.05e49·17-s − 1.67e50·18-s + 5.69e51·19-s − 5.14e52·20-s + 1.69e51·21-s + 9.50e53·22-s − 1.40e55·23-s + 2.97e54·24-s + 9.93e55·25-s − 1.84e56·26-s + 1.46e57·27-s + 2.31e57·28-s + ⋯
L(s)  = 1  + 0.245·2-s − 0.0789·3-s − 0.939·4-s + 1.11·5-s − 0.0193·6-s − 0.0604·7-s − 0.475·8-s − 0.993·9-s + 0.272·10-s + 1.66·11-s + 0.0741·12-s − 0.372·13-s − 0.0148·14-s − 0.0878·15-s + 0.823·16-s − 0.742·17-s − 0.243·18-s + 0.925·19-s − 1.04·20-s + 0.00477·21-s + 0.407·22-s − 0.995·23-s + 0.0375·24-s + 0.240·25-s − 0.0911·26-s + 0.157·27-s + 0.0568·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(82-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+81/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(41.5501\)
Root analytic conductor: \(6.44593\)
Motivic weight: \(81\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :81/2),\ -1)\)

Particular Values

\(L(41)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{83}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 3.81e11T + 2.41e24T^{2} \)
3 \( 1 + 1.66e18T + 4.43e38T^{2} \)
5 \( 1 - 2.26e28T + 4.13e56T^{2} \)
7 \( 1 + 1.01e33T + 2.83e68T^{2} \)
11 \( 1 - 2.49e42T + 2.25e84T^{2} \)
13 \( 1 + 4.84e44T + 1.69e90T^{2} \)
17 \( 1 + 5.05e49T + 4.63e99T^{2} \)
19 \( 1 - 5.69e51T + 3.79e103T^{2} \)
23 \( 1 + 1.40e55T + 1.99e110T^{2} \)
29 \( 1 - 3.83e58T + 2.84e118T^{2} \)
31 \( 1 + 2.12e60T + 6.31e120T^{2} \)
37 \( 1 + 5.54e63T + 1.05e127T^{2} \)
41 \( 1 + 2.25e65T + 4.32e130T^{2} \)
43 \( 1 + 1.09e66T + 2.04e132T^{2} \)
47 \( 1 + 7.15e67T + 2.75e135T^{2} \)
53 \( 1 - 6.78e69T + 4.63e139T^{2} \)
59 \( 1 + 3.00e69T + 2.74e143T^{2} \)
61 \( 1 + 3.61e72T + 4.08e144T^{2} \)
67 \( 1 - 1.41e74T + 8.16e147T^{2} \)
71 \( 1 - 4.91e74T + 8.95e149T^{2} \)
73 \( 1 + 3.62e75T + 8.49e150T^{2} \)
79 \( 1 + 3.03e76T + 5.10e153T^{2} \)
83 \( 1 - 6.46e77T + 2.78e155T^{2} \)
89 \( 1 + 4.68e78T + 7.95e157T^{2} \)
97 \( 1 - 4.00e80T + 8.48e160T^{2} \)
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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

−14.44635039286044105028133076509, −13.66052780842083451381582915790, −11.89601759797250415600425503723, −9.760043758816029885918927792252, −8.761025692396320966779406882470, −6.34536642502006142589396909684, −5.14624879297024743758250587357, −3.50132904840178452344380203754, −1.67715770515130331881599562984, 0, 1.67715770515130331881599562984, 3.50132904840178452344380203754, 5.14624879297024743758250587357, 6.34536642502006142589396909684, 8.761025692396320966779406882470, 9.760043758816029885918927792252, 11.89601759797250415600425503723, 13.66052780842083451381582915790, 14.44635039286044105028133076509

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