Properties

Degree 2
Conductor 2
Sign $1$
Motivic weight 43
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.09e6·2-s − 1.30e10·3-s + 4.39e12·4-s + 1.67e15·5-s + 2.74e16·6-s − 6.70e17·7-s − 9.22e18·8-s − 1.56e20·9-s − 3.50e21·10-s + 2.32e22·11-s − 5.75e22·12-s − 1.22e24·13-s + 1.40e24·14-s − 2.18e25·15-s + 1.93e25·16-s + 3.48e26·17-s + 3.29e26·18-s − 2.92e27·19-s + 7.34e27·20-s + 8.78e27·21-s − 4.88e28·22-s + 2.25e29·23-s + 1.20e29·24-s + 1.65e30·25-s + 2.56e30·26-s + 6.35e30·27-s − 2.95e30·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.722·3-s + 0.5·4-s + 1.56·5-s + 0.510·6-s − 0.453·7-s − 0.353·8-s − 0.478·9-s − 1.10·10-s + 0.948·11-s − 0.361·12-s − 1.37·13-s + 0.321·14-s − 1.13·15-s + 0.250·16-s + 1.22·17-s + 0.338·18-s − 0.939·19-s + 0.783·20-s + 0.327·21-s − 0.670·22-s + 1.19·23-s + 0.255·24-s + 1.45·25-s + 0.970·26-s + 1.06·27-s − 0.226·28-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(43\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2,\ (\ :43/2),\ 1)$
$L(22)$  $\approx$  $1.27587$
$L(\frac12)$  $\approx$  $1.27587$
$L(\frac{45}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + 2.09e6T \)
good3 \( 1 + 1.30e10T + 3.28e20T^{2} \)
5 \( 1 - 1.67e15T + 1.13e30T^{2} \)
7 \( 1 + 6.70e17T + 2.18e36T^{2} \)
11 \( 1 - 2.32e22T + 6.02e44T^{2} \)
13 \( 1 + 1.22e24T + 7.93e47T^{2} \)
17 \( 1 - 3.48e26T + 8.11e52T^{2} \)
19 \( 1 + 2.92e27T + 9.69e54T^{2} \)
23 \( 1 - 2.25e29T + 3.58e58T^{2} \)
29 \( 1 - 4.02e30T + 7.64e62T^{2} \)
31 \( 1 - 1.13e32T + 1.34e64T^{2} \)
37 \( 1 + 7.69e33T + 2.70e67T^{2} \)
41 \( 1 - 3.14e34T + 2.23e69T^{2} \)
43 \( 1 - 1.98e35T + 1.73e70T^{2} \)
47 \( 1 - 1.24e36T + 7.94e71T^{2} \)
53 \( 1 - 2.28e37T + 1.39e74T^{2} \)
59 \( 1 + 5.92e37T + 1.40e76T^{2} \)
61 \( 1 - 1.57e38T + 5.87e76T^{2} \)
67 \( 1 - 1.19e39T + 3.32e78T^{2} \)
71 \( 1 - 7.59e39T + 4.01e79T^{2} \)
73 \( 1 + 1.46e39T + 1.32e80T^{2} \)
79 \( 1 + 2.75e40T + 3.96e81T^{2} \)
83 \( 1 - 2.42e41T + 3.31e82T^{2} \)
89 \( 1 + 5.61e41T + 6.66e83T^{2} \)
97 \( 1 - 3.43e42T + 2.69e85T^{2} \)
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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

−17.33080688422186730237448576095, −16.97653324333278657278071332348, −14.36971796286312416058226775268, −12.28126007326664453601815168481, −10.37818656545802728097423779032, −9.193784287915135026032990493737, −6.68061998450529215251991083419, −5.47618812107195880219590385620, −2.50392942226333406092493706109, −0.878088391737027664620773659112, 0.878088391737027664620773659112, 2.50392942226333406092493706109, 5.47618812107195880219590385620, 6.68061998450529215251991083419, 9.193784287915135026032990493737, 10.37818656545802728097423779032, 12.28126007326664453601815168481, 14.36971796286312416058226775268, 16.97653324333278657278071332348, 17.33080688422186730237448576095

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