Properties

Degree $2$
Conductor $3$
Sign $1$
Motivic weight $41$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.00e6·2-s + 3.48e9·3-s + 1.81e12·4-s − 3.43e14·5-s − 6.98e15·6-s − 2.11e17·7-s + 7.69e17·8-s + 1.21e19·9-s + 6.87e20·10-s − 2.68e21·11-s + 6.32e21·12-s − 4.84e22·13-s + 4.24e23·14-s − 1.19e24·15-s − 5.53e24·16-s − 2.66e25·17-s − 2.43e25·18-s + 4.14e24·19-s − 6.22e26·20-s − 7.38e26·21-s + 5.38e27·22-s − 6.80e27·23-s + 2.68e27·24-s + 7.23e28·25-s + 9.70e28·26-s + 4.23e28·27-s − 3.84e29·28-s + ⋯
L(s)  = 1  − 1.35·2-s + 0.577·3-s + 0.825·4-s − 1.60·5-s − 0.780·6-s − 1.00·7-s + 0.236·8-s + 0.333·9-s + 2.17·10-s − 1.20·11-s + 0.476·12-s − 0.707·13-s + 1.35·14-s − 0.929·15-s − 1.14·16-s − 1.58·17-s − 0.450·18-s + 0.0252·19-s − 1.32·20-s − 0.579·21-s + 1.62·22-s − 0.827·23-s + 0.136·24-s + 1.59·25-s + 0.955·26-s + 0.192·27-s − 0.828·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+41/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $1$
Motivic weight: \(41\)
Character: $\chi_{3} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :41/2),\ 1)\)

Particular Values

\(L(21)\) \(\approx\) \(0.1193251996\)
\(L(\frac12)\) \(\approx\) \(0.1193251996\)
\(L(\frac{43}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3.48e9T \)
good2 \( 1 + 2.00e6T + 2.19e12T^{2} \)
5 \( 1 + 3.43e14T + 4.54e28T^{2} \)
7 \( 1 + 2.11e17T + 4.45e34T^{2} \)
11 \( 1 + 2.68e21T + 4.97e42T^{2} \)
13 \( 1 + 4.84e22T + 4.69e45T^{2} \)
17 \( 1 + 2.66e25T + 2.80e50T^{2} \)
19 \( 1 - 4.14e24T + 2.68e52T^{2} \)
23 \( 1 + 6.80e27T + 6.77e55T^{2} \)
29 \( 1 + 1.53e30T + 9.08e59T^{2} \)
31 \( 1 + 6.13e29T + 1.39e61T^{2} \)
37 \( 1 - 1.47e32T + 1.97e64T^{2} \)
41 \( 1 - 1.43e33T + 1.33e66T^{2} \)
43 \( 1 - 1.93e33T + 9.38e66T^{2} \)
47 \( 1 + 1.54e34T + 3.59e68T^{2} \)
53 \( 1 - 2.77e35T + 4.95e70T^{2} \)
59 \( 1 + 2.82e36T + 4.02e72T^{2} \)
61 \( 1 - 3.86e36T + 1.57e73T^{2} \)
67 \( 1 + 4.59e36T + 7.39e74T^{2} \)
71 \( 1 - 9.12e37T + 7.97e75T^{2} \)
73 \( 1 + 6.94e36T + 2.49e76T^{2} \)
79 \( 1 - 5.97e38T + 6.34e77T^{2} \)
83 \( 1 + 3.36e39T + 4.81e78T^{2} \)
89 \( 1 + 1.32e40T + 8.41e79T^{2} \)
97 \( 1 + 9.71e40T + 2.86e81T^{2} \)
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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

−16.29365181970616152891429923602, −15.41401677996521309696015010742, −12.94943535361367382580416074101, −11.02292721610452548875352312105, −9.487933126176598126347729864258, −8.119062495603604383653260599442, −7.21455050126391732412642137143, −4.16872280881164195584099576642, −2.49429621028374224008217175926, −0.23805643869082347528862747299, 0.23805643869082347528862747299, 2.49429621028374224008217175926, 4.16872280881164195584099576642, 7.21455050126391732412642137143, 8.119062495603604383653260599442, 9.487933126176598126347729864258, 11.02292721610452548875352312105, 12.94943535361367382580416074101, 15.41401677996521309696015010742, 16.29365181970616152891429923602

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