Properties

Degree 4
Conductor $ 2^{2} $
Sign $1$
Motivic weight 43
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.19e6·2-s − 1.29e10·3-s + 1.31e13·4-s − 3.98e14·5-s + 5.44e16·6-s + 1.17e18·7-s − 3.68e19·8-s − 4.86e20·9-s + 1.67e21·10-s − 1.16e22·11-s − 1.71e23·12-s − 1.65e24·13-s − 4.92e24·14-s + 5.17e24·15-s + 9.67e25·16-s + 3.43e26·17-s + 2.04e27·18-s − 5.26e26·19-s − 5.26e27·20-s − 1.52e28·21-s + 4.87e28·22-s + 2.29e29·23-s + 4.78e29·24-s + 1.34e30·25-s + 6.95e30·26-s + 1.05e31·27-s + 1.55e31·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.716·3-s + 3/2·4-s − 0.373·5-s + 1.01·6-s + 0.795·7-s − 1.41·8-s − 1.48·9-s + 0.528·10-s − 0.473·11-s − 1.07·12-s − 1.86·13-s − 1.12·14-s + 0.267·15-s + 5/4·16-s + 1.20·17-s + 2.09·18-s − 0.169·19-s − 0.560·20-s − 0.569·21-s + 0.669·22-s + 1.21·23-s + 1.01·24-s + 1.18·25-s + 2.63·26-s + 1.77·27-s + 1.19·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+43/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(43\)
character  :  induced by $\chi_{2} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 4,\ (\ :43/2, 43/2),\ 1)$
$L(22)$  $\approx$  $0.840322$
$L(\frac12)$  $\approx$  $0.840322$
$L(\frac{45}{2})$   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 4. If $p = 2$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 + p^{21} T )^{2} \)
good3$D_{4}$ \( 1 + 4327210328 p T + 299551664144015314 p^{7} T^{2} + 4327210328 p^{44} T^{3} + p^{86} T^{4} \)
5$D_{4}$ \( 1 + 637860338052 p^{4} T - \)\(24\!\cdots\!58\)\( p^{11} T^{2} + 637860338052 p^{47} T^{3} + p^{86} T^{4} \)
7$D_{4}$ \( 1 - 1174870033543241008 T + \)\(91\!\cdots\!14\)\( p^{3} T^{2} - 1174870033543241008 p^{43} T^{3} + p^{86} T^{4} \)
11$D_{4}$ \( 1 + \)\(10\!\cdots\!16\)\( p T + \)\(32\!\cdots\!86\)\( p^{2} T^{2} + \)\(10\!\cdots\!16\)\( p^{44} T^{3} + p^{86} T^{4} \)
13$D_{4}$ \( 1 + \)\(12\!\cdots\!68\)\( p T + \)\(74\!\cdots\!78\)\( p^{4} T^{2} + \)\(12\!\cdots\!68\)\( p^{44} T^{3} + p^{86} T^{4} \)
17$D_{4}$ \( 1 - \)\(20\!\cdots\!04\)\( p T + \)\(32\!\cdots\!14\)\( p^{3} T^{2} - \)\(20\!\cdots\!04\)\( p^{44} T^{3} + p^{86} T^{4} \)
19$D_{4}$ \( 1 + \)\(52\!\cdots\!00\)\( T + \)\(65\!\cdots\!22\)\( p T^{2} + \)\(52\!\cdots\!00\)\( p^{43} T^{3} + p^{86} T^{4} \)
23$D_{4}$ \( 1 - \)\(22\!\cdots\!36\)\( T + \)\(31\!\cdots\!46\)\( p T^{2} - \)\(22\!\cdots\!36\)\( p^{43} T^{3} + p^{86} T^{4} \)
29$D_{4}$ \( 1 - \)\(11\!\cdots\!80\)\( p T + \)\(19\!\cdots\!58\)\( p^{2} T^{2} - \)\(11\!\cdots\!80\)\( p^{44} T^{3} + p^{86} T^{4} \)
31$D_{4}$ \( 1 - \)\(16\!\cdots\!04\)\( T + \)\(10\!\cdots\!06\)\( p T^{2} - \)\(16\!\cdots\!04\)\( p^{43} T^{3} + p^{86} T^{4} \)
37$D_{4}$ \( 1 + \)\(11\!\cdots\!36\)\( p T + \)\(21\!\cdots\!98\)\( p^{2} T^{2} + \)\(11\!\cdots\!36\)\( p^{44} T^{3} + p^{86} T^{4} \)
41$D_{4}$ \( 1 - \)\(17\!\cdots\!64\)\( p T + \)\(34\!\cdots\!06\)\( p^{2} T^{2} - \)\(17\!\cdots\!64\)\( p^{44} T^{3} + p^{86} T^{4} \)
43$D_{4}$ \( 1 - \)\(23\!\cdots\!36\)\( T + \)\(41\!\cdots\!38\)\( T^{2} - \)\(23\!\cdots\!36\)\( p^{43} T^{3} + p^{86} T^{4} \)
47$D_{4}$ \( 1 + \)\(39\!\cdots\!92\)\( T + \)\(27\!\cdots\!62\)\( T^{2} + \)\(39\!\cdots\!92\)\( p^{43} T^{3} + p^{86} T^{4} \)
53$D_{4}$ \( 1 - \)\(21\!\cdots\!56\)\( T + \)\(24\!\cdots\!38\)\( T^{2} - \)\(21\!\cdots\!56\)\( p^{43} T^{3} + p^{86} T^{4} \)
59$D_{4}$ \( 1 - \)\(67\!\cdots\!40\)\( T + \)\(20\!\cdots\!58\)\( T^{2} - \)\(67\!\cdots\!40\)\( p^{43} T^{3} + p^{86} T^{4} \)
61$D_{4}$ \( 1 - \)\(10\!\cdots\!04\)\( T + \)\(94\!\cdots\!66\)\( T^{2} - \)\(10\!\cdots\!04\)\( p^{43} T^{3} + p^{86} T^{4} \)
67$D_{4}$ \( 1 + \)\(12\!\cdots\!52\)\( T + \)\(36\!\cdots\!02\)\( T^{2} + \)\(12\!\cdots\!52\)\( p^{43} T^{3} + p^{86} T^{4} \)
71$D_{4}$ \( 1 - \)\(10\!\cdots\!64\)\( T + \)\(10\!\cdots\!46\)\( T^{2} - \)\(10\!\cdots\!64\)\( p^{43} T^{3} + p^{86} T^{4} \)
73$D_{4}$ \( 1 + \)\(10\!\cdots\!24\)\( T + \)\(26\!\cdots\!78\)\( T^{2} + \)\(10\!\cdots\!24\)\( p^{43} T^{3} + p^{86} T^{4} \)
79$D_{4}$ \( 1 + \)\(66\!\cdots\!20\)\( T + \)\(90\!\cdots\!78\)\( T^{2} + \)\(66\!\cdots\!20\)\( p^{43} T^{3} + p^{86} T^{4} \)
83$D_{4}$ \( 1 - \)\(54\!\cdots\!56\)\( T + \)\(14\!\cdots\!58\)\( T^{2} - \)\(54\!\cdots\!56\)\( p^{43} T^{3} + p^{86} T^{4} \)
89$D_{4}$ \( 1 - \)\(29\!\cdots\!20\)\( T + \)\(85\!\cdots\!38\)\( T^{2} - \)\(29\!\cdots\!20\)\( p^{43} T^{3} + p^{86} T^{4} \)
97$D_{4}$ \( 1 + \)\(22\!\cdots\!12\)\( T + \)\(34\!\cdots\!82\)\( T^{2} + \)\(22\!\cdots\!12\)\( p^{43} T^{3} + p^{86} T^{4} \)
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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

−17.96160645448336421532711381868, −17.33080688422186730237448576095, −16.97653324333278657278071332348, −16.09370194834217472692218438686, −14.80832250795167993497721456437, −14.36971796286312416058226775268, −12.28126007326664453601815168481, −11.90841038740923063955197932070, −10.94928811191908143949691785540, −10.37818656545802728097423779032, −9.193784287915135026032990493737, −8.295072631434559773077479830200, −7.65723300750753107664232097309, −6.68061998450529215251991083419, −5.47618812107195880219590385620, −4.86185530703986246145813127644, −2.86481725201026409060499933236, −2.50392942226333406092493706109, −0.878088391737027664620773659112, −0.58129219454947557179969273093, 0.58129219454947557179969273093, 0.878088391737027664620773659112, 2.50392942226333406092493706109, 2.86481725201026409060499933236, 4.86185530703986246145813127644, 5.47618812107195880219590385620, 6.68061998450529215251991083419, 7.65723300750753107664232097309, 8.295072631434559773077479830200, 9.193784287915135026032990493737, 10.37818656545802728097423779032, 10.94928811191908143949691785540, 11.90841038740923063955197932070, 12.28126007326664453601815168481, 14.36971796286312416058226775268, 14.80832250795167993497721456437, 16.09370194834217472692218438686, 16.97653324333278657278071332348, 17.33080688422186730237448576095, 17.96160645448336421532711381868

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