Properties

Label 4-87808-1.1-c1e2-0-2
Degree $4$
Conductor $87808$
Sign $1$
Analytic cond. $5.59871$
Root an. cond. $1.53823$
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  + 7-s − 6·9-s + 8·11-s − 6·25-s + 12·29-s − 4·37-s + 8·43-s + 49-s + 12·53-s − 6·63-s + 8·67-s + 16·71-s + 8·77-s − 32·79-s + 27·81-s − 48·99-s + 24·107-s − 20·109-s + 4·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 0.377·7-s − 2·9-s + 2.41·11-s − 6/5·25-s + 2.22·29-s − 0.657·37-s + 1.21·43-s + 1/7·49-s + 1.64·53-s − 0.755·63-s + 0.977·67-s + 1.89·71-s + 0.911·77-s − 3.60·79-s + 3·81-s − 4.82·99-s + 2.32·107-s − 1.91·109-s + 0.376·113-s + 2.36·121-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 + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(87808\)    =    \(2^{8} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(5.59871\)
Root analytic conductor: \(1.53823\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 87808,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.518017094\)
\(L(\frac12)\) \(\approx\) \(1.518017094\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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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

−9.502244133602489113553033388660, −9.117894034638208548514808203742, −8.560607806232820418165457845354, −8.459138850523211127936437645267, −7.80607013998257452679138271336, −6.92383462522053566355627597037, −6.67949194679031572918581545842, −5.84465028770243608573987581113, −5.82613210142107824134972732669, −4.90615311225424089857668766077, −4.18465932715971087267449515138, −3.69860481691342078568866024595, −2.92766858317331148516836536443, −2.14556791802447766709825303506, −0.988078077776060913977316087411, 0.988078077776060913977316087411, 2.14556791802447766709825303506, 2.92766858317331148516836536443, 3.69860481691342078568866024595, 4.18465932715971087267449515138, 4.90615311225424089857668766077, 5.82613210142107824134972732669, 5.84465028770243608573987581113, 6.67949194679031572918581545842, 6.92383462522053566355627597037, 7.80607013998257452679138271336, 8.459138850523211127936437645267, 8.560607806232820418165457845354, 9.117894034638208548514808203742, 9.502244133602489113553033388660

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