Properties

Degree 84
Conductor $ 2^{168} $
Sign $1$
Motivic weight 11
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 1.54e3·4-s − 2·5-s + 4·6-s + 2.18e4·8-s + 2·9-s + 4·10-s − 5.40e5·11-s − 3.08e3·12-s − 2·13-s + 4·15-s + 2.08e5·16-s − 4·17-s − 4·18-s − 1.12e7·19-s − 3.08e3·20-s + 1.08e6·22-s − 4.37e4·24-s + 2·25-s + 4·26-s − 2.22e7·27-s + 7.76e7·29-s − 8·30-s + 3.43e8·31-s + 1.18e8·32-s + 1.08e6·33-s + ⋯
L(s)  = 1  − 0.0441·2-s − 0.00475·3-s + 0.752·4-s − 0.000286·5-s + 0.000210·6-s + 0.235·8-s + 1.12e−5·9-s + 1.26e−5·10-s − 1.01·11-s − 0.00357·12-s − 1.49e − 6·13-s + 1.36e−6·15-s + 0.0497·16-s − 1.04·19-s − 0.000215·20-s + 0.0447·22-s − 0.00112·24-s − 0.298·27-s + 0.703·29-s + 2.15·31-s + 0.626·32-s + 0.00481·33-s + 8.50e−6·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{168}\right)^{s/2} \, \Gamma_{\C}(s)^{42} \, L(s)\cr =\mathstrut & \,\Lambda(12-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{168}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{42} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(84\)
\( N \)  =  \(2^{168}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(11\)
character  :  induced by $\chi_{16} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(84,\ 2^{168} ,\ ( \ : [11/2]^{42} ),\ 1 )$
$L(6)$  $\approx$  $228.043$
$L(\frac12)$  $\approx$  $228.043$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p(T)\) is a polynomial of degree 84. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 83.
$p$$F_p(T)$
bad2 \( 1 + 2T - 1.53e3T^{2} - 2.80e4T^{3} + 2.06e6T^{4} - 3.83e7T^{5} - 3.42e9T^{6} - 4.72e10T^{7} - 2.69e13T^{8} + 1.07e15T^{9} + 7.77e16T^{10} - 1.54e18T^{11} - 1.16e20T^{12} + 4.91e21T^{13} + 1.24e23T^{14} - 8.18e24T^{15} + 1.46e26T^{16} - 1.90e28T^{17} - 7.09e29T^{18} + 7.17e31T^{19} + 8.02e32T^{20} - 1.67e35T^{21} + 1.64e36T^{22} + 3.00e38T^{23} - 6.09e39T^{24} - 3.35e41T^{25} + 5.26e42T^{26} - 6.04e44T^{27} + 1.88e46T^{28} + 1.52e48T^{29} - 7.36e49T^{30} - 2.00e51T^{31} + 2.06e53T^{32} + 5.83e54T^{33} - 3.00e56T^{34}+O(T^{35}) \)
good3 \( 1 + 2T + 2T^{2} + 2.22e7T^{3} - 7.66e10T^{4} + 7.11e12T^{5} + 2.62e14T^{6} - 5.52e18T^{7} + 2.88e21T^{8} - 4.73e23T^{9} - 7.86e25T^{10} + 3.00e29T^{11} - 1.04e32T^{12} + 2.82e33T^{13} + 1.18e37T^{14} - 9.56e39T^{15} + 2.33e42T^{16} + 5.71e44T^{17} - 5.82e47T^{18} + 2.53e50T^{19} + 3.89e52T^{20} - 3.66e55T^{21} + 1.89e58T^{22}+O(T^{23}) \)
5 \( 1 + 2T + 2T^{2} - 1.35e11T^{3} - 3.20e15T^{4} - 2.45e19T^{5} + 9.12e21T^{6} + 5.09e26T^{7} + 1.86e31T^{8} + 6.47e34T^{9} + 2.62e38T^{10} - 4.32e42T^{11} - 6.88e46T^{12} - 3.34e50T^{13} - 2.06e53T^{14} + 7.63e57T^{15}+O(T^{16}) \)
7 \( 1 - 3.72e10T^{2} + 7.00e20T^{4} - 8.84e30T^{6} + 8.43e40T^{8} - 6.48e50T^{10} + 4.18e60T^{12}+O(T^{14}) \)
11 \( 1 + 5.40e5T + 1.46e11T^{2} + 5.02e17T^{3} + 3.45e23T^{4} - 4.33e28T^{5} + 5.22e34T^{6} + 5.56e40T^{7} - 4.84e46T^{8} - 2.67e52T^{9} + 3.46e57T^{10}+O(T^{11}) \)
13 \( 1 + 2T + 2T^{2} - 4.22e17T^{3} + 3.80e24T^{4} - 1.12e30T^{5} + 8.92e34T^{6} + 4.64e42T^{7} + 1.67e48T^{8} - 7.02e54T^{9} - 1.66e60T^{10}+O(T^{11}) \)
17 \( 1 + 4T + 7.22e14T^{2} - 1.68e18T^{3} + 2.62e29T^{4} - 2.72e33T^{5} + 6.39e43T^{6} - 1.41e48T^{7} + 1.17e58T^{8} - 4.22e62T^{9}+O(T^{10}) \)
19 \( 1 + 1.12e7T + 6.37e13T^{2} - 1.87e20T^{3} + 1.10e28T^{4} + 5.10e35T^{5} + 5.07e42T^{6} + 5.66e49T^{7} + 2.34e56T^{8} + 5.18e63T^{9}+O(T^{10}) \)
23 \( 1 - 1.96e16T^{2} + 1.96e32T^{4} - 1.33e48T^{6} + 6.88e63T^{8}+O(T^{9}) \)
29 \( 1 - 7.76e7T + 3.01e15T^{2} - 2.86e24T^{3} + 8.30e31T^{4} + 2.65e40T^{5} + 1.79e48T^{6} + 1.91e56T^{7} - 7.24e64T^{8}+O(T^{9}) \)
31 \( 1 - 3.43e8T + 6.49e17T^{2} - 2.12e26T^{3} + 2.08e35T^{4} - 6.57e43T^{5} + 4.45e52T^{6} - 1.35e61T^{7}+O(T^{8}) \)
37 \( 1 + 5.22e8T + 1.36e17T^{2} - 6.67e25T^{3} - 4.46e34T^{4} + 2.47e43T^{5} + 2.12e52T^{6} + 9.08e60T^{7}+O(T^{8}) \)
41 \( 1 - 1.12e19T^{2} + 6.27e37T^{4} - 2.32e56T^{6}+O(T^{8}) \)
43 \( 1 - 3.82e9T + 7.31e18T^{2} - 1.39e28T^{3} + 2.77e37T^{4} - 4.50e46T^{5} + 6.68e55T^{6} - 1.02e65T^{7}+O(T^{8}) \)
47 \( 1 + 4.58e9T + 5.99e19T^{2} + 2.50e29T^{3} + 1.81e39T^{4} + 6.88e48T^{5} + 3.63e58T^{6} + 1.26e68T^{7}+O(T^{8}) \)
53 \( 1 + 2.10e9T + 2.20e18T^{2} - 5.74e28T^{3} - 4.43e38T^{4} + 6.30e46T^{5} + 2.76e57T^{6}+O(T^{7}) \)
59 \( 1 - 9.55e8T + 4.56e17T^{2} - 1.20e29T^{3} + 1.69e39T^{4} - 6.39e48T^{5} + 1.26e58T^{6}+O(T^{7}) \)
61 \( 1 - 2.15e9T + 2.31e18T^{2} + 5.72e29T^{3} + 4.63e39T^{4} - 4.26e49T^{5} + 2.44e59T^{6}+O(T^{7}) \)
67 \( 1 + 3.18e9T + 5.07e18T^{2} + 1.89e30T^{3} - 5.41e40T^{4} - 1.63e50T^{5} + 1.55e60T^{6}+O(T^{7}) \)
71 \( 1 - 5.54e21T^{2} + 1.50e43T^{4} - 2.64e64T^{6}+O(T^{7}) \)
73 \( 1 - 7.12e21T^{2} + 2.55e43T^{4} - 6.13e64T^{6}+O(T^{7}) \)
79 \( 1 - 4.80e10T + 1.81e22T^{2} - 7.75e32T^{3} + 1.61e44T^{4} - 6.18e54T^{5} + 9.37e65T^{6}+O(T^{7}) \)
83 \( 1 - 5.57e10T + 1.55e21T^{2} - 2.06e32T^{3} + 1.54e43T^{4} - 5.27e53T^{5} + 2.65e64T^{6}+O(T^{7}) \)
89 \( 1 - 6.18e22T^{2} + 1.92e45T^{4} - 4.01e67T^{6}+O(T^{7}) \)
97 \( 1 + 4T + 1.80e23T^{2} + 2.07e33T^{3} + 1.59e46T^{4} + 3.54e56T^{5} + 9.29e68T^{6}+O(T^{7}) \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{84} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.33525155026203846050241903558, −1.30268598160424434064689148834, −1.27223599820369509130419167065, −1.20262085526861008207412444628, −1.16918475730803376399271951682, −1.01294305109356082414147350480, −1.01116125751642421646661300701, −0.995934483925858543306471467300, −0.981528098618663919581607074173, −0.863145047600622248615643477323, −0.827201437368612026926467284042, −0.820884768682237316681174695910, −0.794601337149158505272984983294, −0.74698789235937475480991847399, −0.69392166129695762388087347077, −0.65156579326706514811892427125, −0.57503939181848003014329362622, −0.51840656566161467837431182763, −0.41382110665762507734977227550, −0.40195874175706597785959418791, −0.28392011587841274389626981668, −0.24899681324705062947198842736, −0.17161865364580113431742170449, −0.13994910323790668179551086072, −0.07886282812753708948790843771, 0.07886282812753708948790843771, 0.13994910323790668179551086072, 0.17161865364580113431742170449, 0.24899681324705062947198842736, 0.28392011587841274389626981668, 0.40195874175706597785959418791, 0.41382110665762507734977227550, 0.51840656566161467837431182763, 0.57503939181848003014329362622, 0.65156579326706514811892427125, 0.69392166129695762388087347077, 0.74698789235937475480991847399, 0.794601337149158505272984983294, 0.820884768682237316681174695910, 0.827201437368612026926467284042, 0.863145047600622248615643477323, 0.981528098618663919581607074173, 0.995934483925858543306471467300, 1.01116125751642421646661300701, 1.01294305109356082414147350480, 1.16918475730803376399271951682, 1.20262085526861008207412444628, 1.27223599820369509130419167065, 1.30268598160424434064689148834, 1.33525155026203846050241903558

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

Plot not available for L-functions of degree greater than 10.