Properties

Label 4-840e2-1.1-c1e2-0-7
Degree $4$
Conductor $705600$
Sign $1$
Analytic cond. $44.9896$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 9-s + 4·11-s − 4·19-s − 25-s − 4·29-s − 12·31-s − 20·41-s + 2·45-s − 49-s − 8·55-s + 4·61-s + 28·71-s − 8·79-s + 81-s + 12·89-s + 8·95-s − 4·99-s + 12·101-s + 12·109-s − 10·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯
L(s)  = 1  − 0.894·5-s − 1/3·9-s + 1.20·11-s − 0.917·19-s − 1/5·25-s − 0.742·29-s − 2.15·31-s − 3.12·41-s + 0.298·45-s − 1/7·49-s − 1.07·55-s + 0.512·61-s + 3.32·71-s − 0.900·79-s + 1/9·81-s + 1.27·89-s + 0.820·95-s − 0.402·99-s + 1.19·101-s + 1.14·109-s − 0.909·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(705600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(44.9896\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 705600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.028670841\)
\(L(\frac12)\) \(\approx\) \(1.028670841\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−10.67891358246852338753424523886, −9.822926892149583366672422936997, −9.611823436722503069938489258341, −9.099676340522459889639882847100, −8.669646181919359046844088089535, −8.335427401241195895460483196826, −7.988456332843410587155092074632, −7.26175853751102238744055598102, −7.12164845569631818335312751872, −6.49522545600183232242990988639, −6.24656679780223388853501204015, −5.37199928499264628790821013458, −5.27261076885941236528605010110, −4.43007500352404244440486744602, −4.01954853574863734770663621304, −3.39858133999925403934209529644, −3.37813778600513446651741199482, −2.01192264765215878367684374772, −1.82612035316375870837682414590, −0.48936536022524026085883536692, 0.48936536022524026085883536692, 1.82612035316375870837682414590, 2.01192264765215878367684374772, 3.37813778600513446651741199482, 3.39858133999925403934209529644, 4.01954853574863734770663621304, 4.43007500352404244440486744602, 5.27261076885941236528605010110, 5.37199928499264628790821013458, 6.24656679780223388853501204015, 6.49522545600183232242990988639, 7.12164845569631818335312751872, 7.26175853751102238744055598102, 7.988456332843410587155092074632, 8.335427401241195895460483196826, 8.669646181919359046844088089535, 9.099676340522459889639882847100, 9.611823436722503069938489258341, 9.822926892149583366672422936997, 10.67891358246852338753424523886

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