Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.04e6·2-s + 1.05e10·3-s + 1.09e12·4-s − 2.54e14·5-s + 1.10e16·6-s + 2.96e17·7-s + 1.15e18·8-s + 7.54e19·9-s − 2.66e20·10-s − 1.09e21·11-s + 1.16e22·12-s + 8.09e22·13-s + 3.10e23·14-s − 2.69e24·15-s + 1.20e24·16-s − 5.33e24·17-s + 7.91e25·18-s − 3.50e25·19-s − 2.79e26·20-s + 3.13e27·21-s − 1.14e27·22-s + 6.68e27·23-s + 1.21e28·24-s + 1.91e28·25-s + 8.48e28·26-s + 4.12e29·27-s + 3.25e29·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.75·3-s + 0.5·4-s − 1.19·5-s + 1.23·6-s + 1.40·7-s + 0.353·8-s + 2.06·9-s − 0.843·10-s − 0.491·11-s + 0.876·12-s + 1.18·13-s + 0.991·14-s − 2.08·15-s + 0.250·16-s − 0.318·17-s + 1.46·18-s − 0.214·19-s − 0.596·20-s + 2.45·21-s − 0.347·22-s + 0.812·23-s + 0.619·24-s + 0.421·25-s + 0.835·26-s + 1.87·27-s + 0.701·28-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(41\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2,\ (\ :41/2),\ 1)$
$L(21)$  $\approx$  $5.07957$
$L(\frac12)$  $\approx$  $5.07957$
$L(\frac{43}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - 1.04e6T \)
good3 \( 1 - 1.05e10T + 3.64e19T^{2} \)
5 \( 1 + 2.54e14T + 4.54e28T^{2} \)
7 \( 1 - 2.96e17T + 4.45e34T^{2} \)
11 \( 1 + 1.09e21T + 4.97e42T^{2} \)
13 \( 1 - 8.09e22T + 4.69e45T^{2} \)
17 \( 1 + 5.33e24T + 2.80e50T^{2} \)
19 \( 1 + 3.50e25T + 2.68e52T^{2} \)
23 \( 1 - 6.68e27T + 6.77e55T^{2} \)
29 \( 1 + 1.63e30T + 9.08e59T^{2} \)
31 \( 1 - 2.77e30T + 1.39e61T^{2} \)
37 \( 1 - 4.65e31T + 1.97e64T^{2} \)
41 \( 1 + 8.83e32T + 1.33e66T^{2} \)
43 \( 1 + 3.69e32T + 9.38e66T^{2} \)
47 \( 1 + 1.89e34T + 3.59e68T^{2} \)
53 \( 1 + 2.01e35T + 4.95e70T^{2} \)
59 \( 1 - 5.07e34T + 4.02e72T^{2} \)
61 \( 1 + 6.57e36T + 1.57e73T^{2} \)
67 \( 1 - 5.28e37T + 7.39e74T^{2} \)
71 \( 1 + 1.04e38T + 7.97e75T^{2} \)
73 \( 1 + 2.83e38T + 2.49e76T^{2} \)
79 \( 1 + 6.59e38T + 6.34e77T^{2} \)
83 \( 1 - 1.30e39T + 4.81e78T^{2} \)
89 \( 1 - 2.16e39T + 8.41e79T^{2} \)
97 \( 1 + 8.40e39T + 2.86e81T^{2} \)
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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

−18.78460477151683942421462969779, −15.57017114748410085567426394887, −14.70036525177306077038356057579, −13.28692402703849081191949682565, −11.22235630172916294772901008903, −8.496633578204113835641704222385, −7.59650168024607941287050645931, −4.46147451943213408737526442809, −3.30484653561010155266951809653, −1.68931993327814447396515014980, 1.68931993327814447396515014980, 3.30484653561010155266951809653, 4.46147451943213408737526442809, 7.59650168024607941287050645931, 8.496633578204113835641704222385, 11.22235630172916294772901008903, 13.28692402703849081191949682565, 14.70036525177306077038356057579, 15.57017114748410085567426394887, 18.78460477151683942421462969779

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