Properties

Label 4-840e2-1.1-c1e2-0-58
Degree $4$
Conductor $705600$
Sign $-1$
Analytic cond. $44.9896$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 3·8-s − 9-s − 16-s − 18-s − 25-s + 5·32-s + 36-s − 7·49-s − 50-s + 7·64-s + 16·71-s + 3·72-s + 81-s − 7·98-s + 100-s − 4·113-s − 6·121-s + 127-s − 3·128-s + 131-s + 137-s + 139-s + 16·142-s + 144-s + 149-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1/3·9-s − 1/4·16-s − 0.235·18-s − 1/5·25-s + 0.883·32-s + 1/6·36-s − 49-s − 0.141·50-s + 7/8·64-s + 1.89·71-s + 0.353·72-s + 1/9·81-s − 0.707·98-s + 1/10·100-s − 0.376·113-s − 0.545·121-s + 0.0887·127-s − 0.265·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.34·142-s + 1/12·144-s + 0.0819·149-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: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 705600,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + T^{2} \)
7$C_2$ \( 1 + p T^{2} \)
good11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.106878665527196501488738029768, −7.77221046713237168946486645571, −7.11971522303416405292810269623, −6.54688851024595752035465721198, −6.29855002065771547155921166618, −5.66807838137471404469030056489, −5.31759258948085007443571553816, −4.84516250294896029701802728445, −4.38824987909813670639777395844, −3.75776204854414736459046056773, −3.42673070892602632805527335366, −2.75063685802091318252119195049, −2.16683693140602774553305713214, −1.13451403080775083022194990483, 0, 1.13451403080775083022194990483, 2.16683693140602774553305713214, 2.75063685802091318252119195049, 3.42673070892602632805527335366, 3.75776204854414736459046056773, 4.38824987909813670639777395844, 4.84516250294896029701802728445, 5.31759258948085007443571553816, 5.66807838137471404469030056489, 6.29855002065771547155921166618, 6.54688851024595752035465721198, 7.11971522303416405292810269623, 7.77221046713237168946486645571, 8.106878665527196501488738029768

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