Properties

Degree $4$
Conductor $5308416$
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  − 8·11-s − 4·13-s + 8·23-s + 8·25-s − 16·37-s − 24·47-s + 6·49-s − 16·61-s + 8·71-s − 16·73-s + 24·83-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2.41·11-s − 1.10·13-s + 1.66·23-s + 8/5·25-s − 2.63·37-s − 3.50·47-s + 6/7·49-s − 2.04·61-s + 0.949·71-s − 1.87·73-s + 2.63·83-s − 2.68·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5308416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4617232442\)
\(L(\frac12)\) \(\approx\) \(0.4617232442\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 64 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{2} \)
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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

−9.283076350652407326201495881785, −8.650370595617985789180098837586, −8.469740669072292414134061951165, −8.104566962140610598435152614983, −7.50861622863333305290099105873, −7.34150211670380816578073557738, −7.06911132119200124551994124987, −6.40355310819957039614955280145, −6.26777613181024333962562965269, −5.30477956450098129495367544897, −5.11265373585282500258274198123, −5.00009400216933859675251187869, −4.75394705023948498917572289366, −3.84043259140866042327596939201, −3.26131021009050861886019039244, −2.83941373417171295977862401120, −2.70281061195281806164723263700, −1.92829210324510103586957248605, −1.31870632846192158168050730455, −0.23054144447577228874751391699, 0.23054144447577228874751391699, 1.31870632846192158168050730455, 1.92829210324510103586957248605, 2.70281061195281806164723263700, 2.83941373417171295977862401120, 3.26131021009050861886019039244, 3.84043259140866042327596939201, 4.75394705023948498917572289366, 5.00009400216933859675251187869, 5.11265373585282500258274198123, 5.30477956450098129495367544897, 6.26777613181024333962562965269, 6.40355310819957039614955280145, 7.06911132119200124551994124987, 7.34150211670380816578073557738, 7.50861622863333305290099105873, 8.104566962140610598435152614983, 8.469740669072292414134061951165, 8.650370595617985789180098837586, 9.283076350652407326201495881785

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