Properties

Label 4-7800e2-1.1-c1e2-0-3
Degree $4$
Conductor $60840000$
Sign $1$
Analytic cond. $3879.21$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 7-s + 3·9-s − 7·11-s − 2·13-s + 17-s − 2·19-s + 2·21-s + 7·23-s + 4·27-s − 4·29-s − 6·31-s − 14·33-s + 37-s − 4·39-s − 7·41-s + 4·43-s − 6·47-s − 9·49-s + 2·51-s − 3·53-s − 4·57-s − 16·59-s − 9·61-s + 3·63-s + 12·67-s + 14·69-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 9-s − 2.11·11-s − 0.554·13-s + 0.242·17-s − 0.458·19-s + 0.436·21-s + 1.45·23-s + 0.769·27-s − 0.742·29-s − 1.07·31-s − 2.43·33-s + 0.164·37-s − 0.640·39-s − 1.09·41-s + 0.609·43-s − 0.875·47-s − 9/7·49-s + 0.280·51-s − 0.412·53-s − 0.529·57-s − 2.08·59-s − 1.15·61-s + 0.377·63-s + 1.46·67-s + 1.68·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(60840000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3879.21\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 60840000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T - 32 T^{2} - p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 7 T + 56 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T + 104 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 9 T + 104 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 3 T + 106 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 142 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 54 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 17 T + 212 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 15 T + 144 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
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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

−7.68272922787228454529493909979, −7.58641825541479827540430036621, −7.09872079578051009473735180149, −6.84900451905353027028379191787, −6.26615140737730023667786304311, −6.03673040190530908472335232884, −5.29781195617123054747216811270, −5.23149284877696839981335537443, −4.89380252116516263380057536750, −4.58969256206134114059974237496, −4.08567840253363864292127311848, −3.55133010117424190709096055351, −3.19076589916989981262717284582, −2.95162915122110165806360903424, −2.43213820424258063885599855822, −2.23533871389667684011556812253, −1.51581194973951569214074106347, −1.31763615702739426816392650246, 0, 0, 1.31763615702739426816392650246, 1.51581194973951569214074106347, 2.23533871389667684011556812253, 2.43213820424258063885599855822, 2.95162915122110165806360903424, 3.19076589916989981262717284582, 3.55133010117424190709096055351, 4.08567840253363864292127311848, 4.58969256206134114059974237496, 4.89380252116516263380057536750, 5.23149284877696839981335537443, 5.29781195617123054747216811270, 6.03673040190530908472335232884, 6.26615140737730023667786304311, 6.84900451905353027028379191787, 7.09872079578051009473735180149, 7.58641825541479827540430036621, 7.68272922787228454529493909979

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