Properties

Label 2-1-1.1-c99-0-0
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $62.0676$
Root an. cond. $7.87830$
Motivic weight $99$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.53e15·2-s + 1.09e23·3-s + 1.73e30·4-s − 1.42e34·5-s − 1.68e38·6-s − 4.25e41·7-s − 1.69e45·8-s − 1.59e47·9-s + 2.19e49·10-s − 5.04e51·11-s + 1.90e53·12-s − 1.33e55·13-s + 6.54e56·14-s − 1.55e57·15-s + 1.51e60·16-s + 8.54e60·17-s + 2.45e62·18-s − 3.57e63·19-s − 2.47e64·20-s − 4.65e64·21-s + 7.76e66·22-s − 1.32e67·23-s − 1.85e68·24-s − 1.37e69·25-s + 2.05e70·26-s − 3.63e70·27-s − 7.37e71·28-s + ⋯
L(s)  = 1  − 1.93·2-s + 0.264·3-s + 2.73·4-s − 0.358·5-s − 0.510·6-s − 0.625·7-s − 3.36·8-s − 0.930·9-s + 0.693·10-s − 1.42·11-s + 0.723·12-s − 0.965·13-s + 1.20·14-s − 0.0947·15-s + 3.76·16-s + 1.05·17-s + 1.79·18-s − 1.80·19-s − 0.981·20-s − 0.165·21-s + 2.75·22-s − 0.521·23-s − 0.887·24-s − 0.871·25-s + 1.86·26-s − 0.509·27-s − 1.71·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(100-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+99/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(62.0676\)
Root analytic conductor: \(7.87830\)
Motivic weight: \(99\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :99/2),\ 1)\)

Particular Values

\(L(50)\) \(\approx\) \(0.003122643029\)
\(L(\frac12)\) \(\approx\) \(0.003122643029\)
\(L(\frac{101}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 1.53e15T + 6.33e29T^{2} \)
3 \( 1 - 1.09e23T + 1.71e47T^{2} \)
5 \( 1 + 1.42e34T + 1.57e69T^{2} \)
7 \( 1 + 4.25e41T + 4.62e83T^{2} \)
11 \( 1 + 5.04e51T + 1.25e103T^{2} \)
13 \( 1 + 1.33e55T + 1.90e110T^{2} \)
17 \( 1 - 8.54e60T + 6.52e121T^{2} \)
19 \( 1 + 3.57e63T + 3.95e126T^{2} \)
23 \( 1 + 1.32e67T + 6.47e134T^{2} \)
29 \( 1 + 1.57e71T + 5.98e144T^{2} \)
31 \( 1 + 3.89e73T + 4.41e147T^{2} \)
37 \( 1 + 1.80e77T + 1.78e155T^{2} \)
41 \( 1 - 2.13e79T + 4.63e159T^{2} \)
43 \( 1 - 8.43e79T + 5.16e161T^{2} \)
47 \( 1 - 2.19e82T + 3.44e165T^{2} \)
53 \( 1 + 2.60e85T + 5.05e170T^{2} \)
59 \( 1 - 3.91e87T + 2.06e175T^{2} \)
61 \( 1 + 3.67e88T + 5.59e176T^{2} \)
67 \( 1 + 3.53e90T + 6.04e180T^{2} \)
71 \( 1 - 1.43e91T + 1.88e183T^{2} \)
73 \( 1 + 5.96e91T + 2.94e184T^{2} \)
79 \( 1 + 1.81e93T + 7.32e187T^{2} \)
83 \( 1 + 6.69e93T + 9.74e189T^{2} \)
89 \( 1 + 5.60e95T + 9.76e192T^{2} \)
97 \( 1 + 2.13e98T + 4.90e196T^{2} \)
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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

−15.01146275052369743077562627650, −12.32112540394735054472525046695, −10.79369323364500928403001998356, −9.726817868569553135269251489866, −8.356708014895255088732055096616, −7.51695230002812897172164015992, −5.92501523376097362495176638710, −3.02403823756730519695357521273, −2.07466531707269433382459856444, −0.03599348156476027294440974586, 0.03599348156476027294440974586, 2.07466531707269433382459856444, 3.02403823756730519695357521273, 5.92501523376097362495176638710, 7.51695230002812897172164015992, 8.356708014895255088732055096616, 9.726817868569553135269251489866, 10.79369323364500928403001998356, 12.32112540394735054472525046695, 15.01146275052369743077562627650

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