Properties

Degree 4
Conductor $ 2^{8} $
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  − 56·3-s + 7.86e3·5-s − 9.10e4·7-s + 9.45e4·9-s − 1.59e5·11-s + 1.05e6·13-s − 4.40e5·15-s + 1.43e6·17-s + 2.18e7·19-s + 5.09e6·21-s − 3.58e7·23-s + 1.30e7·25-s − 2.03e7·27-s − 2.28e8·29-s − 6.47e7·31-s + 8.90e6·33-s − 7.16e8·35-s + 7.55e7·37-s − 5.88e7·39-s + 1.20e9·41-s + 4.55e7·43-s + 7.43e8·45-s + 1.22e9·47-s + 3.05e9·49-s − 8.01e7·51-s − 3.80e9·53-s − 1.25e9·55-s + ⋯
L(s)  = 1  − 0.133·3-s + 1.12·5-s − 2.04·7-s + 0.533·9-s − 0.297·11-s + 0.784·13-s − 0.149·15-s + 0.244·17-s + 2.02·19-s + 0.272·21-s − 1.16·23-s + 0.267·25-s − 0.272·27-s − 2.07·29-s − 0.406·31-s + 0.0396·33-s − 2.30·35-s + 0.179·37-s − 0.104·39-s + 1.61·41-s + 0.0472·43-s + 0.600·45-s + 0.781·47-s + 1.54·49-s − 0.0325·51-s − 1.25·53-s − 0.335·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(256\)    =    \(2^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(11\)
character  :  induced by $\chi_{16} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 256,\ (\ :11/2, 11/2),\ 1)$
$L(6)$  $\approx$  $1.96925$
$L(\frac12)$  $\approx$  $1.96925$
$L(\frac{13}{2})$   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 4. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
good3$D_{4}$ \( 1 + 56 T - 10154 p^{2} T^{2} + 56 p^{11} T^{3} + p^{22} T^{4} \)
5$D_{4}$ \( 1 - 7868 T + 9768358 p T^{2} - 7868 p^{11} T^{3} + p^{22} T^{4} \)
7$D_{4}$ \( 1 + 13008 p T + 106936526 p^{2} T^{2} + 13008 p^{12} T^{3} + p^{22} T^{4} \)
11$D_{4}$ \( 1 + 159080 T + 362536063286 T^{2} + 159080 p^{11} T^{3} + p^{22} T^{4} \)
13$D_{4}$ \( 1 - 1050476 T + 3458956762062 T^{2} - 1050476 p^{11} T^{3} + p^{22} T^{4} \)
17$D_{4}$ \( 1 - 1430884 T + 16748384809766 T^{2} - 1430884 p^{11} T^{3} + p^{22} T^{4} \)
19$D_{4}$ \( 1 - 21866600 T + 343123710088422 T^{2} - 21866600 p^{11} T^{3} + p^{22} T^{4} \)
23$D_{4}$ \( 1 + 35806736 T + 1952928132896462 T^{2} + 35806736 p^{11} T^{3} + p^{22} T^{4} \)
29$D_{4}$ \( 1 + 228827700 T + 35907977791027054 T^{2} + 228827700 p^{11} T^{3} + p^{22} T^{4} \)
31$D_{4}$ \( 1 + 64722112 T + 51498184379920062 T^{2} + 64722112 p^{11} T^{3} + p^{22} T^{4} \)
37$D_{4}$ \( 1 - 75558780 T + 354292341022528382 T^{2} - 75558780 p^{11} T^{3} + p^{22} T^{4} \)
41$D_{4}$ \( 1 - 1201214196 T + 1274442257818453270 T^{2} - 1201214196 p^{11} T^{3} + p^{22} T^{4} \)
43$D_{4}$ \( 1 - 45519832 T + 42144714696540642 p T^{2} - 45519832 p^{11} T^{3} + p^{22} T^{4} \)
47$D_{4}$ \( 1 - 1229079264 T + 5024390553242310430 T^{2} - 1229079264 p^{11} T^{3} + p^{22} T^{4} \)
53$D_{4}$ \( 1 + 3808549924 T + 16648234587904299038 T^{2} + 3808549924 p^{11} T^{3} + p^{22} T^{4} \)
59$D_{4}$ \( 1 - 6012926584 T + 61066799326040273366 T^{2} - 6012926584 p^{11} T^{3} + p^{22} T^{4} \)
61$D_{4}$ \( 1 - 9789792908 T + \)\(10\!\cdots\!42\)\( T^{2} - 9789792908 p^{11} T^{3} + p^{22} T^{4} \)
67$D_{4}$ \( 1 + 14703095224 T + 95414710392374126214 T^{2} + 14703095224 p^{11} T^{3} + p^{22} T^{4} \)
71$D_{4}$ \( 1 + 4319991088 T + \)\(28\!\cdots\!82\)\( T^{2} + 4319991088 p^{11} T^{3} + p^{22} T^{4} \)
73$D_{4}$ \( 1 - 11055639476 T + \)\(65\!\cdots\!62\)\( T^{2} - 11055639476 p^{11} T^{3} + p^{22} T^{4} \)
79$D_{4}$ \( 1 + 51957623264 T + \)\(14\!\cdots\!78\)\( T^{2} + 51957623264 p^{11} T^{3} + p^{22} T^{4} \)
83$D_{4}$ \( 1 - 108227975912 T + \)\(54\!\cdots\!06\)\( T^{2} - 108227975912 p^{11} T^{3} + p^{22} T^{4} \)
89$D_{4}$ \( 1 - 71188291860 T + \)\(31\!\cdots\!82\)\( T^{2} - 71188291860 p^{11} T^{3} + p^{22} T^{4} \)
97$D_{4}$ \( 1 + 1699807676 T + \)\(12\!\cdots\!50\)\( T^{2} + 1699807676 p^{11} T^{3} + p^{22} T^{4} \)
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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

−16.31116200825258545253202759527, −16.27275781335032660734699212824, −15.77208591190569214929515396042, −14.66617974980762707720612933035, −13.78348259345486604673612963891, −13.27298414774464976662333642755, −12.90621233315029395497971725832, −12.06742731671037754752253715619, −11.04009576396169403919370865238, −10.09885135781673194795594083335, −9.503765499426568174650429211507, −9.356921975928151278472735814093, −7.74152778914749856244192914165, −6.91628249184811814931534013839, −5.87315045012548812973032793679, −5.66823903622263311409643752644, −3.86952645670578587420913653289, −3.08831314804859228294228836739, −1.85827559864606508570080203569, −0.61854315435287421392366405863, 0.61854315435287421392366405863, 1.85827559864606508570080203569, 3.08831314804859228294228836739, 3.86952645670578587420913653289, 5.66823903622263311409643752644, 5.87315045012548812973032793679, 6.91628249184811814931534013839, 7.74152778914749856244192914165, 9.356921975928151278472735814093, 9.503765499426568174650429211507, 10.09885135781673194795594083335, 11.04009576396169403919370865238, 12.06742731671037754752253715619, 12.90621233315029395497971725832, 13.27298414774464976662333642755, 13.78348259345486604673612963891, 14.66617974980762707720612933035, 15.77208591190569214929515396042, 16.27275781335032660734699212824, 16.31116200825258545253202759527

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