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  + 9.49e5·2-s + 3.48e9·3-s − 1.29e12·4-s − 1.14e14·5-s + 3.30e15·6-s − 7.22e16·7-s − 3.31e18·8-s + 1.21e19·9-s − 1.08e20·10-s + 2.08e21·11-s − 4.52e21·12-s + 1.02e23·13-s − 6.85e22·14-s − 3.99e23·15-s − 2.96e23·16-s + 1.36e25·17-s + 1.15e25·18-s + 1.23e26·19-s + 1.48e26·20-s − 2.51e26·21-s + 1.97e27·22-s − 4.97e27·23-s − 1.15e28·24-s − 3.23e28·25-s + 9.73e28·26-s + 4.23e28·27-s + 9.37e28·28-s + ⋯
L(s)  = 1  + 0.640·2-s + 0.577·3-s − 0.590·4-s − 0.537·5-s + 0.369·6-s − 0.342·7-s − 1.01·8-s + 0.333·9-s − 0.344·10-s + 0.934·11-s − 0.340·12-s + 1.49·13-s − 0.219·14-s − 0.310·15-s − 0.0612·16-s + 0.812·17-s + 0.213·18-s + 0.754·19-s + 0.317·20-s − 0.197·21-s + 0.598·22-s − 0.604·23-s − 0.587·24-s − 0.710·25-s + 0.958·26-s + 0.192·27-s + 0.201·28-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

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(41\)
character  :  $\chi_{3} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3,\ (\ :41/2),\ 1)\)
\(L(21)\)  \(\approx\)  \(2.715961259\)
\(L(\frac12)\)  \(\approx\)  \(2.715961259\)
\(L(\frac{43}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3$,\(F_p(T)\) is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - 3.48e9T \)
good2 \( 1 - 9.49e5T + 2.19e12T^{2} \)
5 \( 1 + 1.14e14T + 4.54e28T^{2} \)
7 \( 1 + 7.22e16T + 4.45e34T^{2} \)
11 \( 1 - 2.08e21T + 4.97e42T^{2} \)
13 \( 1 - 1.02e23T + 4.69e45T^{2} \)
17 \( 1 - 1.36e25T + 2.80e50T^{2} \)
19 \( 1 - 1.23e26T + 2.68e52T^{2} \)
23 \( 1 + 4.97e27T + 6.77e55T^{2} \)
29 \( 1 - 7.01e29T + 9.08e59T^{2} \)
31 \( 1 - 1.80e30T + 1.39e61T^{2} \)
37 \( 1 - 2.23e32T + 1.97e64T^{2} \)
41 \( 1 - 1.70e33T + 1.33e66T^{2} \)
43 \( 1 + 2.72e33T + 9.38e66T^{2} \)
47 \( 1 + 4.84e33T + 3.59e68T^{2} \)
53 \( 1 + 5.39e33T + 4.95e70T^{2} \)
59 \( 1 - 3.17e36T + 4.02e72T^{2} \)
61 \( 1 + 1.06e36T + 1.57e73T^{2} \)
67 \( 1 + 3.93e37T + 7.39e74T^{2} \)
71 \( 1 + 1.01e38T + 7.97e75T^{2} \)
73 \( 1 - 2.40e38T + 2.49e76T^{2} \)
79 \( 1 - 3.87e38T + 6.34e77T^{2} \)
83 \( 1 - 7.28e38T + 4.81e78T^{2} \)
89 \( 1 + 1.16e40T + 8.41e79T^{2} \)
97 \( 1 - 9.69e40T + 2.86e81T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.01011278978685888613057125820, −14.49087059391990460760191877346, −13.36384551678528203776537808837, −11.83821856368051836119808538702, −9.544162973716166835633178740028, −8.168099141132219016509541559910, −6.09121522419422348833758769714, −4.16279728815865988996188367555, −3.27823885956949930655935892256, −0.967755440504959470005255867116, 0.967755440504959470005255867116, 3.27823885956949930655935892256, 4.16279728815865988996188367555, 6.09121522419422348833758769714, 8.168099141132219016509541559910, 9.544162973716166835633178740028, 11.83821856368051836119808538702, 13.36384551678528203776537808837, 14.49087059391990460760191877346, 16.01011278978685888613057125820

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