Properties

Degree $4$
Conductor $65536$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·7-s + 2·9-s − 4·17-s + 8·23-s + 6·25-s + 12·41-s + 16·47-s + 34·49-s − 16·63-s + 24·71-s − 28·73-s + 16·79-s − 5·81-s + 4·89-s − 4·97-s − 8·103-s + 4·113-s + 32·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + ⋯
L(s)  = 1  − 3.02·7-s + 2/3·9-s − 0.970·17-s + 1.66·23-s + 6/5·25-s + 1.87·41-s + 2.33·47-s + 34/7·49-s − 2.01·63-s + 2.84·71-s − 3.27·73-s + 1.80·79-s − 5/9·81-s + 0.423·89-s − 0.406·97-s − 0.788·103-s + 0.376·113-s + 2.93·119-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.646·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65536\)    =    \(2^{16}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{256} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.943494\)
\(L(\frac12)\) \(\approx\) \(0.943494\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.46448586956374693128491949424, −12.01993603461779646561261081518, −11.05675814829446524247374047496, −10.82899110790509360137108028749, −10.31055245362243336087047752436, −9.776138170650733784260645661580, −9.423884709811879950950983404496, −8.870868915425085523508767953007, −8.851882489860000138251155333534, −7.56515021467950703892918779460, −7.20786807714381711280566941520, −6.66103041189581929330436903728, −6.44976683667966550556096328535, −5.83522495830210206987552686355, −5.08762159732160482445369213130, −4.21895121524273615476920088960, −3.73740289547189202194632871913, −2.90739036926013457245631156495, −2.60239244415137240943522716091, −0.77554292563920746309047607113, 0.77554292563920746309047607113, 2.60239244415137240943522716091, 2.90739036926013457245631156495, 3.73740289547189202194632871913, 4.21895121524273615476920088960, 5.08762159732160482445369213130, 5.83522495830210206987552686355, 6.44976683667966550556096328535, 6.66103041189581929330436903728, 7.20786807714381711280566941520, 7.56515021467950703892918779460, 8.851882489860000138251155333534, 8.870868915425085523508767953007, 9.423884709811879950950983404496, 9.776138170650733784260645661580, 10.31055245362243336087047752436, 10.82899110790509360137108028749, 11.05675814829446524247374047496, 12.01993603461779646561261081518, 12.46448586956374693128491949424

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