Properties

Degree $2$
Conductor $1$
Sign $-1$
Motivic weight $93$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.55e14·2-s + 1.43e22·3-s + 1.43e28·4-s − 3.85e32·5-s + 2.23e36·6-s − 1.93e39·7-s + 6.89e41·8-s − 2.95e43·9-s − 6.00e46·10-s + 4.19e48·11-s + 2.05e50·12-s − 6.72e51·13-s − 3.01e53·14-s − 5.53e54·15-s − 3.46e55·16-s + 7.90e56·17-s − 4.60e57·18-s − 1.84e59·19-s − 5.53e60·20-s − 2.78e61·21-s + 6.52e62·22-s − 3.52e63·23-s + 9.89e63·24-s + 4.79e64·25-s − 1.04e66·26-s − 3.80e66·27-s − 2.77e67·28-s + ⋯
L(s)  = 1  + 1.56·2-s + 0.935·3-s + 1.44·4-s − 1.21·5-s + 1.46·6-s − 0.977·7-s + 0.699·8-s − 0.125·9-s − 1.89·10-s + 1.57·11-s + 1.35·12-s − 1.07·13-s − 1.52·14-s − 1.13·15-s − 0.352·16-s + 0.480·17-s − 0.196·18-s − 0.637·19-s − 1.75·20-s − 0.913·21-s + 2.46·22-s − 1.68·23-s + 0.654·24-s + 0.474·25-s − 1.67·26-s − 1.05·27-s − 1.41·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(94-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+93/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Motivic weight: \(93\)
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :93/2),\ -1)\)

Particular Values

\(L(47)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{95}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 1.55e14T + 9.90e27T^{2} \)
3 \( 1 - 1.43e22T + 2.35e44T^{2} \)
5 \( 1 + 3.85e32T + 1.00e65T^{2} \)
7 \( 1 + 1.93e39T + 3.92e78T^{2} \)
11 \( 1 - 4.19e48T + 7.07e96T^{2} \)
13 \( 1 + 6.72e51T + 3.95e103T^{2} \)
17 \( 1 - 7.90e56T + 2.70e114T^{2} \)
19 \( 1 + 1.84e59T + 8.39e118T^{2} \)
23 \( 1 + 3.52e63T + 4.37e126T^{2} \)
29 \( 1 - 6.08e66T + 1.00e136T^{2} \)
31 \( 1 + 3.87e69T + 4.97e138T^{2} \)
37 \( 1 - 9.82e72T + 6.96e145T^{2} \)
41 \( 1 - 1.01e75T + 9.74e149T^{2} \)
43 \( 1 - 9.34e75T + 8.17e151T^{2} \)
47 \( 1 + 5.01e76T + 3.19e155T^{2} \)
53 \( 1 - 1.92e80T + 2.27e160T^{2} \)
59 \( 1 + 2.20e81T + 4.88e164T^{2} \)
61 \( 1 + 3.04e82T + 1.08e166T^{2} \)
67 \( 1 + 8.28e84T + 6.68e169T^{2} \)
71 \( 1 - 1.80e86T + 1.46e172T^{2} \)
73 \( 1 + 2.33e86T + 1.94e173T^{2} \)
79 \( 1 + 4.15e87T + 3.01e176T^{2} \)
83 \( 1 - 5.26e88T + 2.98e178T^{2} \)
89 \( 1 - 3.32e90T + 1.96e181T^{2} \)
97 \( 1 - 2.28e92T + 5.88e184T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.23744535615545311274684423027, −12.60629857805690976263939524483, −11.68510079557927615475191460248, −9.256526175280520049805026727069, −7.51240547974848520329960058488, −6.08620355525781336302059505149, −4.09951250907526273424642340885, −3.60911542290329273613954173090, −2.37613694941362268153927827413, 0, 2.37613694941362268153927827413, 3.60911542290329273613954173090, 4.09951250907526273424642340885, 6.08620355525781336302059505149, 7.51240547974848520329960058488, 9.256526175280520049805026727069, 11.68510079557927615475191460248, 12.60629857805690976263939524483, 14.23744535615545311274684423027

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