Properties

Degree 4
Conductor $ 2^{2} $
Sign $1$
Motivic weight 41
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.09e6·2-s + 8.86e9·3-s + 3.29e12·4-s + 9.75e13·5-s + 1.85e16·6-s + 2.17e17·7-s + 4.61e18·8-s + 2.37e19·9-s + 2.04e20·10-s + 1.10e20·11-s + 2.92e22·12-s + 1.73e23·13-s + 4.55e23·14-s + 8.65e23·15-s + 6.04e24·16-s − 1.02e25·17-s + 4.98e25·18-s + 2.06e26·19-s + 3.21e26·20-s + 1.92e27·21-s + 2.31e26·22-s + 1.49e28·23-s + 4.08e28·24-s + 8.05e27·25-s + 3.63e29·26-s + 4.87e28·27-s + 7.16e29·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.46·3-s + 3/2·4-s + 0.457·5-s + 2.07·6-s + 1.02·7-s + 1.41·8-s + 0.652·9-s + 0.647·10-s + 0.0495·11-s + 2.20·12-s + 2.52·13-s + 1.45·14-s + 0.671·15-s + 5/4·16-s − 0.613·17-s + 0.922·18-s + 1.25·19-s + 0.686·20-s + 1.50·21-s + 0.0700·22-s + 1.81·23-s + 2.07·24-s + 0.177·25-s + 3.57·26-s + 0.221·27-s + 1.54·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+41/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(41\)
character  :  induced by $\chi_{2} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 4,\ (\ :41/2, 41/2),\ 1)$
$L(21)$  $\approx$  $18.1697$
$L(\frac12)$  $\approx$  $18.1697$
$L(\frac{43}{2})$   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 4. If $p = 2$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 - p^{20} T )^{2} \)
good3$D_{4}$ \( 1 - 984816392 p^{2} T + 927466955744998 p^{10} T^{2} - 984816392 p^{43} T^{3} + p^{82} T^{4} \)
5$D_{4}$ \( 1 - 19519836865116 p T + \)\(47\!\cdots\!46\)\( p^{5} T^{2} - 19519836865116 p^{42} T^{3} + p^{82} T^{4} \)
7$D_{4}$ \( 1 - 4430751068838544 p^{2} T + \)\(79\!\cdots\!86\)\( p^{7} T^{2} - 4430751068838544 p^{43} T^{3} + p^{82} T^{4} \)
11$D_{4}$ \( 1 - 10054043771358142344 p T + \)\(64\!\cdots\!06\)\( p^{3} T^{2} - 10054043771358142344 p^{42} T^{3} + p^{82} T^{4} \)
13$D_{4}$ \( 1 - \)\(13\!\cdots\!36\)\( p T + \)\(76\!\cdots\!06\)\( p^{3} T^{2} - \)\(13\!\cdots\!36\)\( p^{42} T^{3} + p^{82} T^{4} \)
17$D_{4}$ \( 1 + \)\(10\!\cdots\!04\)\( T + \)\(34\!\cdots\!14\)\( p T^{2} + \)\(10\!\cdots\!04\)\( p^{41} T^{3} + p^{82} T^{4} \)
19$D_{4}$ \( 1 - \)\(10\!\cdots\!00\)\( p T + \)\(65\!\cdots\!82\)\( p^{3} T^{2} - \)\(10\!\cdots\!00\)\( p^{42} T^{3} + p^{82} T^{4} \)
23$D_{4}$ \( 1 - \)\(14\!\cdots\!68\)\( T + \)\(82\!\cdots\!74\)\( p T^{2} - \)\(14\!\cdots\!68\)\( p^{41} T^{3} + p^{82} T^{4} \)
29$D_{4}$ \( 1 + \)\(47\!\cdots\!80\)\( p T + \)\(47\!\cdots\!02\)\( p T^{2} + \)\(47\!\cdots\!80\)\( p^{42} T^{3} + p^{82} T^{4} \)
31$D_{4}$ \( 1 + \)\(11\!\cdots\!36\)\( p T + \)\(10\!\cdots\!66\)\( p^{2} T^{2} + \)\(11\!\cdots\!36\)\( p^{42} T^{3} + p^{82} T^{4} \)
37$D_{4}$ \( 1 - \)\(32\!\cdots\!28\)\( p T + \)\(31\!\cdots\!42\)\( p^{2} T^{2} - \)\(32\!\cdots\!28\)\( p^{42} T^{3} + p^{82} T^{4} \)
41$D_{4}$ \( 1 - \)\(12\!\cdots\!64\)\( T + \)\(75\!\cdots\!06\)\( T^{2} - \)\(12\!\cdots\!64\)\( p^{41} T^{3} + p^{82} T^{4} \)
43$D_{4}$ \( 1 + \)\(60\!\cdots\!52\)\( T + \)\(20\!\cdots\!62\)\( T^{2} + \)\(60\!\cdots\!52\)\( p^{41} T^{3} + p^{82} T^{4} \)
47$D_{4}$ \( 1 + \)\(25\!\cdots\!64\)\( T + \)\(84\!\cdots\!18\)\( T^{2} + \)\(25\!\cdots\!64\)\( p^{41} T^{3} + p^{82} T^{4} \)
53$D_{4}$ \( 1 + \)\(30\!\cdots\!52\)\( T + \)\(11\!\cdots\!82\)\( T^{2} + \)\(30\!\cdots\!52\)\( p^{41} T^{3} + p^{82} T^{4} \)
59$D_{4}$ \( 1 + \)\(32\!\cdots\!40\)\( T + \)\(78\!\cdots\!18\)\( T^{2} + \)\(32\!\cdots\!40\)\( p^{41} T^{3} + p^{82} T^{4} \)
61$D_{4}$ \( 1 + \)\(44\!\cdots\!36\)\( T + \)\(17\!\cdots\!46\)\( T^{2} + \)\(44\!\cdots\!36\)\( p^{41} T^{3} + p^{82} T^{4} \)
67$D_{4}$ \( 1 - \)\(55\!\cdots\!56\)\( T + \)\(15\!\cdots\!18\)\( T^{2} - \)\(55\!\cdots\!56\)\( p^{41} T^{3} + p^{82} T^{4} \)
71$D_{4}$ \( 1 + \)\(48\!\cdots\!16\)\( T + \)\(10\!\cdots\!06\)\( T^{2} + \)\(48\!\cdots\!16\)\( p^{41} T^{3} + p^{82} T^{4} \)
73$D_{4}$ \( 1 + \)\(48\!\cdots\!12\)\( T + \)\(10\!\cdots\!82\)\( T^{2} + \)\(48\!\cdots\!12\)\( p^{41} T^{3} + p^{82} T^{4} \)
79$D_{4}$ \( 1 + \)\(90\!\cdots\!80\)\( T + \)\(14\!\cdots\!58\)\( T^{2} + \)\(90\!\cdots\!80\)\( p^{41} T^{3} + p^{82} T^{4} \)
83$D_{4}$ \( 1 - \)\(12\!\cdots\!68\)\( T + \)\(95\!\cdots\!22\)\( T^{2} - \)\(12\!\cdots\!68\)\( p^{41} T^{3} + p^{82} T^{4} \)
89$D_{4}$ \( 1 - \)\(26\!\cdots\!80\)\( T + \)\(16\!\cdots\!78\)\( T^{2} - \)\(26\!\cdots\!80\)\( p^{41} T^{3} + p^{82} T^{4} \)
97$D_{4}$ \( 1 + \)\(31\!\cdots\!04\)\( T + \)\(59\!\cdots\!98\)\( T^{2} + \)\(31\!\cdots\!04\)\( p^{41} T^{3} + p^{82} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.78460477151683942421462969779, −17.96160461071481857497799814239, −16.50444697546084319802645030174, −15.57017114748410085567426394887, −14.70036525177306077038356057579, −14.26588627843624105406328000972, −13.28692402703849081191949682565, −13.16628907179816938592848982938, −11.22235630172916294772901008903, −11.12914078299579263768529450063, −9.266134554387374088129282103860, −8.496633578204113835641704222385, −7.59650168024607941287050645931, −6.39130410049944954266522517283, −5.42902154586998137447859334147, −4.46147451943213408737526442809, −3.30484653561010155266951809653, −3.08688946578043229440874758303, −1.68931993327814447396515014980, −1.38155187090491114534548989819, 1.38155187090491114534548989819, 1.68931993327814447396515014980, 3.08688946578043229440874758303, 3.30484653561010155266951809653, 4.46147451943213408737526442809, 5.42902154586998137447859334147, 6.39130410049944954266522517283, 7.59650168024607941287050645931, 8.496633578204113835641704222385, 9.266134554387374088129282103860, 11.12914078299579263768529450063, 11.22235630172916294772901008903, 13.16628907179816938592848982938, 13.28692402703849081191949682565, 14.26588627843624105406328000972, 14.70036525177306077038356057579, 15.57017114748410085567426394887, 16.50444697546084319802645030174, 17.96160461071481857497799814239, 18.78460477151683942421462969779

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