Properties

Label 4-3240e2-1.1-c1e2-0-35
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $669.336$
Root an. cond. $5.08640$
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  − 2·5-s + 2·7-s − 4·17-s − 4·19-s − 10·23-s + 3·25-s + 6·29-s + 12·31-s − 4·35-s − 12·37-s − 10·41-s + 4·43-s − 14·47-s − 5·49-s − 4·53-s − 12·59-s − 6·61-s − 2·67-s + 8·73-s − 4·79-s − 6·83-s + 8·85-s − 14·89-s + 8·95-s + 4·97-s − 4·101-s + 20·103-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s − 0.970·17-s − 0.917·19-s − 2.08·23-s + 3/5·25-s + 1.11·29-s + 2.15·31-s − 0.676·35-s − 1.97·37-s − 1.56·41-s + 0.609·43-s − 2.04·47-s − 5/7·49-s − 0.549·53-s − 1.56·59-s − 0.768·61-s − 0.244·67-s + 0.936·73-s − 0.450·79-s − 0.658·83-s + 0.867·85-s − 1.48·89-s + 0.820·95-s + 0.406·97-s − 0.398·101-s + 1.97·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(669.336\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3240} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 10 T + 65 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 10 T + 83 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 6 T + 169 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 14 T + 131 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.321954330565190670546770237618, −8.269388673049869867898993227212, −7.69799642451249789294204202953, −7.62620403829308572052042681192, −6.82501363862108435672879883240, −6.58729483272453982115367604885, −6.31437347731658171487128154550, −6.03260523213820954563537902392, −5.08397461220960252514209057093, −5.04924821575820287588360703708, −4.47352745114991563769336165146, −4.37672227492823335198910506060, −3.74112630462186499594093168576, −3.39645753062151709406053779632, −2.76685277250510753791582560652, −2.35525540850745789774399018372, −1.62008066329913966364782445589, −1.39616136812889307768968616451, 0, 0, 1.39616136812889307768968616451, 1.62008066329913966364782445589, 2.35525540850745789774399018372, 2.76685277250510753791582560652, 3.39645753062151709406053779632, 3.74112630462186499594093168576, 4.37672227492823335198910506060, 4.47352745114991563769336165146, 5.04924821575820287588360703708, 5.08397461220960252514209057093, 6.03260523213820954563537902392, 6.31437347731658171487128154550, 6.58729483272453982115367604885, 6.82501363862108435672879883240, 7.62620403829308572052042681192, 7.69799642451249789294204202953, 8.269388673049869867898993227212, 8.321954330565190670546770237618

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