Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.13e4·2-s + 3.82e9·3-s − 2.19e12·4-s + 9.10e13·5-s − 3.10e14·6-s − 1.69e16·7-s + 3.57e17·8-s − 2.18e19·9-s − 7.40e18·10-s − 3.48e21·11-s − 8.37e21·12-s − 1.03e23·13-s + 1.38e21·14-s + 3.48e23·15-s + 4.79e24·16-s + 1.01e25·17-s + 1.77e24·18-s − 1.72e26·19-s − 1.99e26·20-s − 6.49e25·21-s + 2.83e26·22-s + 1.09e28·23-s + 1.36e27·24-s − 3.71e28·25-s + 8.39e27·26-s − 2.22e29·27-s + 3.72e28·28-s + ⋯
L(s)  = 1  − 0.0548·2-s + 0.632·3-s − 0.996·4-s + 0.426·5-s − 0.0347·6-s − 0.0804·7-s + 0.109·8-s − 0.599·9-s − 0.0234·10-s − 1.56·11-s − 0.630·12-s − 1.50·13-s + 0.00441·14-s + 0.270·15-s + 0.990·16-s + 0.606·17-s + 0.0328·18-s − 1.05·19-s − 0.425·20-s − 0.0509·21-s + 0.0857·22-s + 1.33·23-s + 0.0693·24-s − 0.817·25-s + 0.0825·26-s − 1.01·27-s + 0.0802·28-s + ⋯

Functional equation

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

Invariants

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

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, \(F_p\) is a polynomial of degree 2.
$p$$F_p$
good2 \( 1 + 8.13e4T + 2.19e12T^{2} \)
3 \( 1 - 3.82e9T + 3.64e19T^{2} \)
5 \( 1 - 9.10e13T + 4.54e28T^{2} \)
7 \( 1 + 1.69e16T + 4.45e34T^{2} \)
11 \( 1 + 3.48e21T + 4.97e42T^{2} \)
13 \( 1 + 1.03e23T + 4.69e45T^{2} \)
17 \( 1 - 1.01e25T + 2.80e50T^{2} \)
19 \( 1 + 1.72e26T + 2.68e52T^{2} \)
23 \( 1 - 1.09e28T + 6.77e55T^{2} \)
29 \( 1 - 1.22e30T + 9.08e59T^{2} \)
31 \( 1 - 5.57e29T + 1.39e61T^{2} \)
37 \( 1 - 1.14e32T + 1.97e64T^{2} \)
41 \( 1 + 6.17e32T + 1.33e66T^{2} \)
43 \( 1 + 1.51e32T + 9.38e66T^{2} \)
47 \( 1 + 1.94e34T + 3.59e68T^{2} \)
53 \( 1 + 1.38e35T + 4.95e70T^{2} \)
59 \( 1 - 2.19e36T + 4.02e72T^{2} \)
61 \( 1 - 2.83e36T + 1.57e73T^{2} \)
67 \( 1 - 2.71e37T + 7.39e74T^{2} \)
71 \( 1 - 4.91e37T + 7.97e75T^{2} \)
73 \( 1 + 2.29e38T + 2.49e76T^{2} \)
79 \( 1 - 1.69e38T + 6.34e77T^{2} \)
83 \( 1 + 3.52e39T + 4.81e78T^{2} \)
89 \( 1 - 6.37e39T + 8.41e79T^{2} \)
97 \( 1 - 3.79e40T + 2.86e81T^{2} \)
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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

−21.35115002574498219475902348316, −19.21081244188731159242930442973, −17.38772926926616988119887193712, −14.61898485256595024349163130251, −13.05589820423207455440841210847, −9.886708198321619448957698846734, −8.152569295302234085979200853491, −5.09857215754134194960555388559, −2.71348491106079875612499419113, 0, 2.71348491106079875612499419113, 5.09857215754134194960555388559, 8.152569295302234085979200853491, 9.886708198321619448957698846734, 13.05589820423207455440841210847, 14.61898485256595024349163130251, 17.38772926926616988119887193712, 19.21081244188731159242930442973, 21.35115002574498219475902348316

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