Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 105
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.55e15·2-s − 1.37e25·3-s + 5.07e31·4-s − 2.80e36·5-s + 1.31e41·6-s − 2.78e44·7-s − 9.70e46·8-s + 6.48e49·9-s + 2.67e52·10-s + 6.08e53·11-s − 6.99e56·12-s − 5.49e58·13-s + 2.65e60·14-s + 3.86e61·15-s − 1.13e63·16-s − 2.90e64·17-s − 6.19e65·18-s − 1.56e67·19-s − 1.42e68·20-s + 3.83e69·21-s − 5.81e69·22-s + 5.72e71·23-s + 1.33e72·24-s − 1.68e73·25-s + 5.24e74·26-s + 8.33e74·27-s − 1.41e76·28-s + ⋯
L(s)  = 1  − 1.50·2-s − 1.23·3-s + 1.25·4-s − 0.564·5-s + 1.84·6-s − 1.19·7-s − 0.375·8-s + 0.517·9-s + 0.846·10-s + 0.129·11-s − 1.54·12-s − 1.80·13-s + 1.78·14-s + 0.694·15-s − 0.686·16-s − 0.731·17-s − 0.776·18-s − 1.14·19-s − 0.705·20-s + 1.46·21-s − 0.193·22-s + 1.84·23-s + 0.462·24-s − 0.681·25-s + 2.71·26-s + 0.594·27-s − 1.49·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(106-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+52.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(105\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :105/2),\ -1)\)
\(L(53)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{107}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 2.
$p$$F_p(T)$
good2 \( 1 + 9.55e15T + 4.05e31T^{2} \)
3 \( 1 + 1.37e25T + 1.25e50T^{2} \)
5 \( 1 + 2.80e36T + 2.46e73T^{2} \)
7 \( 1 + 2.78e44T + 5.43e88T^{2} \)
11 \( 1 - 6.08e53T + 2.21e109T^{2} \)
13 \( 1 + 5.49e58T + 9.20e116T^{2} \)
17 \( 1 + 2.90e64T + 1.57e129T^{2} \)
19 \( 1 + 1.56e67T + 1.85e134T^{2} \)
23 \( 1 - 5.72e71T + 9.58e142T^{2} \)
29 \( 1 + 1.76e76T + 3.56e153T^{2} \)
31 \( 1 - 1.61e78T + 3.91e156T^{2} \)
37 \( 1 - 1.96e82T + 4.58e164T^{2} \)
41 \( 1 - 1.17e83T + 2.19e169T^{2} \)
43 \( 1 - 1.98e84T + 3.26e171T^{2} \)
47 \( 1 - 1.10e88T + 3.71e175T^{2} \)
53 \( 1 + 7.34e89T + 1.11e181T^{2} \)
59 \( 1 + 1.37e93T + 8.69e185T^{2} \)
61 \( 1 - 1.74e93T + 2.88e187T^{2} \)
67 \( 1 - 1.25e96T + 5.46e191T^{2} \)
71 \( 1 + 1.81e97T + 2.41e194T^{2} \)
73 \( 1 + 2.37e97T + 4.45e195T^{2} \)
79 \( 1 - 8.68e98T + 1.78e199T^{2} \)
83 \( 1 - 2.69e100T + 3.18e201T^{2} \)
89 \( 1 + 5.78e101T + 4.85e204T^{2} \)
97 \( 1 - 1.53e103T + 4.08e208T^{2} \)
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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

−12.66323796617456848954104266175, −11.37408163513349698196670288558, −10.24190745145928612759084338581, −9.083051712698212924923901800579, −7.36176336770378457356853210865, −6.41211190653736602546487150668, −4.62962452377337361497565598717, −2.52156215037672634036692424523, −0.65376610677889205136415830235, 0, 0.65376610677889205136415830235, 2.52156215037672634036692424523, 4.62962452377337361497565598717, 6.41211190653736602546487150668, 7.36176336770378457356853210865, 9.083051712698212924923901800579, 10.24190745145928612759084338581, 11.37408163513349698196670288558, 12.66323796617456848954104266175

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