Properties

Label 4-7800e2-1.1-c1e2-0-5
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 7-s + 3·9-s − 5·11-s + 2·13-s − 5·17-s − 4·19-s + 2·21-s − 7·23-s + 4·27-s + 6·29-s − 6·31-s − 10·33-s + 7·37-s + 4·39-s + 15·41-s − 8·43-s − 10·47-s − 3·49-s − 10·51-s − 15·53-s − 8·57-s − 4·59-s − 61-s + 3·63-s + 8·67-s − 14·69-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 9-s − 1.50·11-s + 0.554·13-s − 1.21·17-s − 0.917·19-s + 0.436·21-s − 1.45·23-s + 0.769·27-s + 1.11·29-s − 1.07·31-s − 1.74·33-s + 1.15·37-s + 0.640·39-s + 2.34·41-s − 1.21·43-s − 1.45·47-s − 3/7·49-s − 1.40·51-s − 2.06·53-s − 1.05·57-s − 0.520·59-s − 0.128·61-s + 0.377·63-s + 0.977·67-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: Trivial
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$D_{4}$ \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 7 T + 76 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 15 T + 128 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 15 T + 152 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + T + 112 T^{2} + p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + T - 114 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 150 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 3 T + 88 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 17 T + 174 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
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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

−7.71098470068746665119787970138, −7.67785063904964523830487751811, −7.00720693043923507696612725962, −6.63783635077317268159108038936, −6.33451669882169697129633635754, −6.10187293409555604159188222360, −5.52946084825338551462627779345, −5.20913974332270176626736752648, −4.71053643156127050637246655074, −4.44486068928596405058379669717, −4.04825501854536297769230575033, −3.87619878443687571066490410839, −3.09524265628747451708048227255, −2.90866040352754123176957515802, −2.38899228883228091557027120002, −2.25772694655059107006728264401, −1.52289531914570518966651386584, −1.33660567640394530148269003333, 0, 0, 1.33660567640394530148269003333, 1.52289531914570518966651386584, 2.25772694655059107006728264401, 2.38899228883228091557027120002, 2.90866040352754123176957515802, 3.09524265628747451708048227255, 3.87619878443687571066490410839, 4.04825501854536297769230575033, 4.44486068928596405058379669717, 4.71053643156127050637246655074, 5.20913974332270176626736752648, 5.52946084825338551462627779345, 6.10187293409555604159188222360, 6.33451669882169697129633635754, 6.63783635077317268159108038936, 7.00720693043923507696612725962, 7.67785063904964523830487751811, 7.71098470068746665119787970138

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