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  + 2.58e6·2-s + 3.48e9·3-s + 4.47e12·4-s + 2.55e14·5-s + 9.00e15·6-s + 1.98e17·7-s + 5.88e18·8-s + 1.21e19·9-s + 6.60e20·10-s − 2.76e21·11-s + 1.56e22·12-s − 8.56e22·13-s + 5.14e23·14-s + 8.91e23·15-s + 5.36e24·16-s − 3.38e24·17-s + 3.14e25·18-s + 2.99e26·19-s + 1.14e27·20-s + 6.93e26·21-s − 7.14e27·22-s − 1.40e28·23-s + 2.05e28·24-s + 1.98e28·25-s − 2.21e29·26-s + 4.23e28·27-s + 8.90e29·28-s + ⋯
L(s)  = 1  + 1.74·2-s + 0.577·3-s + 2.03·4-s + 1.19·5-s + 1.00·6-s + 0.942·7-s + 1.80·8-s + 0.333·9-s + 2.08·10-s − 1.23·11-s + 1.17·12-s − 1.24·13-s + 1.64·14-s + 0.691·15-s + 1.10·16-s − 0.202·17-s + 0.580·18-s + 1.82·19-s + 2.43·20-s + 0.544·21-s − 2.15·22-s − 1.71·23-s + 1.04·24-s + 0.436·25-s − 2.17·26-s + 0.192·27-s + 1.91·28-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

\(L(21)\) \(\approx\) \(8.326711406\)
\(L(\frac12)\) \(\approx\) \(8.326711406\)
\(L(\frac{43}{2})\) 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 \( 1 - 2.58e6T + 2.19e12T^{2} \)
5 \( 1 - 2.55e14T + 4.54e28T^{2} \)
7 \( 1 - 1.98e17T + 4.45e34T^{2} \)
11 \( 1 + 2.76e21T + 4.97e42T^{2} \)
13 \( 1 + 8.56e22T + 4.69e45T^{2} \)
17 \( 1 + 3.38e24T + 2.80e50T^{2} \)
19 \( 1 - 2.99e26T + 2.68e52T^{2} \)
23 \( 1 + 1.40e28T + 6.77e55T^{2} \)
29 \( 1 + 4.94e28T + 9.08e59T^{2} \)
31 \( 1 - 5.76e30T + 1.39e61T^{2} \)
37 \( 1 + 1.16e32T + 1.97e64T^{2} \)
41 \( 1 - 1.48e32T + 1.33e66T^{2} \)
43 \( 1 - 2.64e32T + 9.38e66T^{2} \)
47 \( 1 - 2.16e34T + 3.59e68T^{2} \)
53 \( 1 - 5.57e34T + 4.95e70T^{2} \)
59 \( 1 + 2.05e36T + 4.02e72T^{2} \)
61 \( 1 - 1.01e36T + 1.57e73T^{2} \)
67 \( 1 + 2.42e37T + 7.39e74T^{2} \)
71 \( 1 - 5.77e36T + 7.97e75T^{2} \)
73 \( 1 + 1.22e38T + 2.49e76T^{2} \)
79 \( 1 + 7.70e38T + 6.34e77T^{2} \)
83 \( 1 - 1.70e39T + 4.81e78T^{2} \)
89 \( 1 + 4.08e39T + 8.41e79T^{2} \)
97 \( 1 - 1.60e40T + 2.86e81T^{2} \)
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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

−15.66299243622715711507632580677, −14.21702509205311263202458446500, −13.60348642219944607458708691625, −12.04123026317238516471208353394, −10.08139757531473401389038677824, −7.57692701435250980805046229326, −5.67464253258325347012894904810, −4.71538140726844446009909166281, −2.80124302747572001597600018569, −1.91314548178943544225527390400, 1.91314548178943544225527390400, 2.80124302747572001597600018569, 4.71538140726844446009909166281, 5.67464253258325347012894904810, 7.57692701435250980805046229326, 10.08139757531473401389038677824, 12.04123026317238516471208353394, 13.60348642219944607458708691625, 14.21702509205311263202458446500, 15.66299243622715711507632580677

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