Properties

Label 2-2-1.1-c81-0-0
Degree $2$
Conductor $2$
Sign $1$
Analytic cond. $83.1002$
Root an. cond. $9.11593$
Motivic weight $81$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.09e12·2-s − 1.09e19·3-s + 1.20e24·4-s − 1.35e28·5-s − 1.20e31·6-s − 2.01e34·7-s + 1.32e36·8-s − 3.23e38·9-s − 1.49e40·10-s + 1.34e41·11-s − 1.32e43·12-s − 1.41e45·13-s − 2.21e46·14-s + 1.48e47·15-s + 1.46e48·16-s − 6.93e49·17-s − 3.55e50·18-s + 2.85e51·19-s − 1.64e52·20-s + 2.20e53·21-s + 1.47e53·22-s + 1.73e55·23-s − 1.45e55·24-s − 2.28e56·25-s − 1.55e57·26-s + 8.39e57·27-s − 2.43e58·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.519·3-s + 0.5·4-s − 0.668·5-s − 0.367·6-s − 1.19·7-s + 0.353·8-s − 0.729·9-s − 0.472·10-s + 0.0893·11-s − 0.259·12-s − 1.08·13-s − 0.843·14-s + 0.347·15-s + 0.250·16-s − 1.01·17-s − 0.516·18-s + 0.463·19-s − 0.334·20-s + 0.620·21-s + 0.0631·22-s + 1.23·23-s − 0.183·24-s − 0.553·25-s − 0.766·26-s + 0.898·27-s − 0.596·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2\)
Sign: $1$
Analytic conductor: \(83.1002\)
Root analytic conductor: \(9.11593\)
Motivic weight: \(81\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2,\ (\ :81/2),\ 1)\)

Particular Values

\(L(41)\) \(\approx\) \(0.9933188689\)
\(L(\frac12)\) \(\approx\) \(0.9933188689\)
\(L(\frac{83}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.09e12T \)
good3 \( 1 + 1.09e19T + 4.43e38T^{2} \)
5 \( 1 + 1.35e28T + 4.13e56T^{2} \)
7 \( 1 + 2.01e34T + 2.83e68T^{2} \)
11 \( 1 - 1.34e41T + 2.25e84T^{2} \)
13 \( 1 + 1.41e45T + 1.69e90T^{2} \)
17 \( 1 + 6.93e49T + 4.63e99T^{2} \)
19 \( 1 - 2.85e51T + 3.79e103T^{2} \)
23 \( 1 - 1.73e55T + 1.99e110T^{2} \)
29 \( 1 + 1.94e59T + 2.84e118T^{2} \)
31 \( 1 + 8.81e59T + 6.31e120T^{2} \)
37 \( 1 - 2.77e63T + 1.05e127T^{2} \)
41 \( 1 + 2.02e65T + 4.32e130T^{2} \)
43 \( 1 - 2.10e66T + 2.04e132T^{2} \)
47 \( 1 - 2.81e67T + 2.75e135T^{2} \)
53 \( 1 + 7.63e69T + 4.63e139T^{2} \)
59 \( 1 - 4.52e71T + 2.74e143T^{2} \)
61 \( 1 - 4.55e71T + 4.08e144T^{2} \)
67 \( 1 + 1.17e74T + 8.16e147T^{2} \)
71 \( 1 - 7.88e74T + 8.95e149T^{2} \)
73 \( 1 + 5.80e73T + 8.49e150T^{2} \)
79 \( 1 - 7.16e76T + 5.10e153T^{2} \)
83 \( 1 - 4.36e77T + 2.78e155T^{2} \)
89 \( 1 - 1.73e79T + 7.95e157T^{2} \)
97 \( 1 - 4.37e80T + 8.48e160T^{2} \)
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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

−13.20787315390451749296021973500, −12.06949900410206718081277174120, −11.00165953419685141596102635182, −9.307857905810617658682698436958, −7.39560665790946365817304013983, −6.24623737766837934786737635905, −4.99223663401325734079075783047, −3.60782561744738757361252162056, −2.50877781832259357027743113090, −0.43404206843309624699846110995, 0.43404206843309624699846110995, 2.50877781832259357027743113090, 3.60782561744738757361252162056, 4.99223663401325734079075783047, 6.24623737766837934786737635905, 7.39560665790946365817304013983, 9.307857905810617658682698436958, 11.00165953419685141596102635182, 12.06949900410206718081277174120, 13.20787315390451749296021973500

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