Properties

Label 4-2200e2-1.1-c1e2-0-14
Degree $4$
Conductor $4840000$
Sign $1$
Analytic cond. $308.602$
Root an. cond. $4.19131$
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  − 3-s − 5·7-s − 9-s − 2·11-s − 6·13-s + 3·17-s + 7·19-s + 5·21-s + 6·23-s + 13·29-s − 7·31-s + 2·33-s − 19·37-s + 6·39-s − 8·41-s − 2·43-s − 2·47-s + 9·49-s − 3·51-s − 7·53-s − 7·57-s − 6·59-s − 21·61-s + 5·63-s − 6·69-s − 5·71-s + 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.88·7-s − 1/3·9-s − 0.603·11-s − 1.66·13-s + 0.727·17-s + 1.60·19-s + 1.09·21-s + 1.25·23-s + 2.41·29-s − 1.25·31-s + 0.348·33-s − 3.12·37-s + 0.960·39-s − 1.24·41-s − 0.304·43-s − 0.291·47-s + 9/7·49-s − 0.420·51-s − 0.961·53-s − 0.927·57-s − 0.781·59-s − 2.68·61-s + 0.629·63-s − 0.722·69-s − 0.593·71-s + 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4840000\)    =    \(2^{6} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(308.602\)
Root analytic conductor: \(4.19131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4840000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_4$ \( 1 - 7 T + 46 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 19 T + 160 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T - 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 7 T + 80 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 21 T + 228 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 7 T + 152 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 18 T + 258 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
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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

−8.810396538863175355179999383484, −8.785402291776056964809231708808, −7.84392484829148481377827725106, −7.67172082986766673362073512417, −7.28109951824093198299008720832, −6.84440514355326304137414267431, −6.53999919205739018185548424832, −6.22890009525466020544546426678, −5.64324103213377772433090446760, −5.16380139545867065441953244621, −4.91771582944466859018289888002, −4.84019275934143573144119954468, −3.65716234947209555278738095568, −3.37702356766540203005667795978, −2.95911103601770103329387322429, −2.83740340563170843539856462838, −1.88546454629694874648105451316, −1.14086041607664668277828186316, 0, 0, 1.14086041607664668277828186316, 1.88546454629694874648105451316, 2.83740340563170843539856462838, 2.95911103601770103329387322429, 3.37702356766540203005667795978, 3.65716234947209555278738095568, 4.84019275934143573144119954468, 4.91771582944466859018289888002, 5.16380139545867065441953244621, 5.64324103213377772433090446760, 6.22890009525466020544546426678, 6.53999919205739018185548424832, 6.84440514355326304137414267431, 7.28109951824093198299008720832, 7.67172082986766673362073512417, 7.84392484829148481377827725106, 8.785402291776056964809231708808, 8.810396538863175355179999383484

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