Properties

Degree $4$
Conductor $9144576$
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  + 4·7-s − 4·19-s + 7·25-s + 12·29-s + 4·31-s + 2·37-s − 6·47-s + 9·49-s + 24·53-s + 6·59-s − 18·83-s + 16·103-s − 14·109-s + 12·113-s + 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.917·19-s + 7/5·25-s + 2.22·29-s + 0.718·31-s + 0.328·37-s − 0.875·47-s + 9/7·49-s + 3.29·53-s + 0.781·59-s − 1.97·83-s + 1.57·103-s − 1.34·109-s + 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.047429592\)
\(L(\frac12)\) \(\approx\) \(4.047429592\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 155 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 182 T^{2} + p^{2} T^{4} \)
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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

−8.615569725576451338330027133995, −8.596145739783728379344149036936, −8.213700704646330522583304468171, −8.034925581622747721991614425366, −7.24614113952464245359521577162, −7.12082006532309916508943525772, −6.73961271961020402772990681626, −6.30170367510470301474186080147, −5.79484833374628755163853077930, −5.47163910060977024061727019801, −4.81809910941579823533122901722, −4.80627462284735285995418671014, −4.20492015467822639094068765075, −4.07162418257711142966903336517, −3.12895080752269676522959116115, −2.87684242128108222421340472720, −2.24569575228573954634634820170, −1.89020282111464465092488994864, −1.05047054358958422598057850152, −0.76973689203848763629691502914, 0.76973689203848763629691502914, 1.05047054358958422598057850152, 1.89020282111464465092488994864, 2.24569575228573954634634820170, 2.87684242128108222421340472720, 3.12895080752269676522959116115, 4.07162418257711142966903336517, 4.20492015467822639094068765075, 4.80627462284735285995418671014, 4.81809910941579823533122901722, 5.47163910060977024061727019801, 5.79484833374628755163853077930, 6.30170367510470301474186080147, 6.73961271961020402772990681626, 7.12082006532309916508943525772, 7.24614113952464245359521577162, 8.034925581622747721991614425366, 8.213700704646330522583304468171, 8.596145739783728379344149036936, 8.615569725576451338330027133995

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