Properties

Degree $2$
Conductor $3$
Sign $-1$
Motivic weight $43$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.22e6·2-s + 1.04e10·3-s − 7.30e12·4-s − 8.84e14·5-s − 1.27e16·6-s + 1.48e18·7-s + 1.96e19·8-s + 1.09e20·9-s + 1.07e21·10-s + 1.46e21·11-s − 7.64e22·12-s − 4.36e23·13-s − 1.80e24·14-s − 9.24e24·15-s + 4.02e25·16-s + 3.51e26·17-s − 1.33e26·18-s − 1.50e27·19-s + 6.45e27·20-s + 1.54e28·21-s − 1.78e27·22-s − 1.10e29·23-s + 2.05e29·24-s − 3.55e29·25-s + 5.33e29·26-s + 1.14e30·27-s − 1.08e31·28-s + ⋯
L(s)  = 1  − 0.411·2-s + 0.577·3-s − 0.830·4-s − 0.829·5-s − 0.237·6-s + 1.00·7-s + 0.753·8-s + 0.333·9-s + 0.341·10-s + 0.0596·11-s − 0.479·12-s − 0.489·13-s − 0.412·14-s − 0.478·15-s + 0.519·16-s + 1.23·17-s − 0.137·18-s − 0.481·19-s + 0.688·20-s + 0.578·21-s − 0.0245·22-s − 0.582·23-s + 0.435·24-s − 0.312·25-s + 0.201·26-s + 0.192·27-s − 0.831·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-1$
Motivic weight: \(43\)
Character: $\chi_{3} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3,\ (\ :43/2),\ -1)\)

Particular Values

\(L(22)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{45}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.04e10T \)
good2 \( 1 + 1.22e6T + 8.79e12T^{2} \)
5 \( 1 + 8.84e14T + 1.13e30T^{2} \)
7 \( 1 - 1.48e18T + 2.18e36T^{2} \)
11 \( 1 - 1.46e21T + 6.02e44T^{2} \)
13 \( 1 + 4.36e23T + 7.93e47T^{2} \)
17 \( 1 - 3.51e26T + 8.11e52T^{2} \)
19 \( 1 + 1.50e27T + 9.69e54T^{2} \)
23 \( 1 + 1.10e29T + 3.58e58T^{2} \)
29 \( 1 + 3.00e31T + 7.64e62T^{2} \)
31 \( 1 - 1.83e32T + 1.34e64T^{2} \)
37 \( 1 + 9.66e33T + 2.70e67T^{2} \)
41 \( 1 + 5.73e34T + 2.23e69T^{2} \)
43 \( 1 + 1.24e35T + 1.73e70T^{2} \)
47 \( 1 + 6.52e35T + 7.94e71T^{2} \)
53 \( 1 - 2.03e37T + 1.39e74T^{2} \)
59 \( 1 + 7.05e37T + 1.40e76T^{2} \)
61 \( 1 + 4.16e38T + 5.87e76T^{2} \)
67 \( 1 + 1.46e39T + 3.32e78T^{2} \)
71 \( 1 - 6.58e39T + 4.01e79T^{2} \)
73 \( 1 - 8.76e39T + 1.32e80T^{2} \)
79 \( 1 + 3.63e39T + 3.96e81T^{2} \)
83 \( 1 + 2.43e41T + 3.31e82T^{2} \)
89 \( 1 - 6.12e41T + 6.66e83T^{2} \)
97 \( 1 - 8.54e41T + 2.69e85T^{2} \)
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

−15.07369273742691870053081870967, −13.86114570105810332994260477440, −12.00930564359947662396997850230, −10.11157787148284646165772175715, −8.478431318381445473061522867460, −7.65736410488286676200001533813, −4.93026348265395101803088471747, −3.66289620441689658929329871020, −1.56479521037483426254455630649, 0, 1.56479521037483426254455630649, 3.66289620441689658929329871020, 4.93026348265395101803088471747, 7.65736410488286676200001533813, 8.478431318381445473061522867460, 10.11157787148284646165772175715, 12.00930564359947662396997850230, 13.86114570105810332994260477440, 15.07369273742691870053081870967

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