Properties

Degree $4$
Conductor $7139584$
Sign $1$
Motivic weight $1$
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 − 6·5-s − 3·7-s − 2·9-s + 9·13-s + 6·15-s + 17-s − 4·19-s + 3·21-s + 23-s + 17·25-s + 2·27-s + 8·29-s − 6·31-s + 18·35-s + 8·37-s − 9·39-s + 2·41-s − 12·43-s + 12·45-s + 4·47-s − 4·49-s − 51-s − 2·53-s + 4·57-s + 6·59-s − 8·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 2.68·5-s − 1.13·7-s − 2/3·9-s + 2.49·13-s + 1.54·15-s + 0.242·17-s − 0.917·19-s + 0.654·21-s + 0.208·23-s + 17/5·25-s + 0.384·27-s + 1.48·29-s − 1.07·31-s + 3.04·35-s + 1.31·37-s − 1.44·39-s + 0.312·41-s − 1.82·43-s + 1.78·45-s + 0.583·47-s − 4/7·49-s − 0.140·51-s − 0.274·53-s + 0.529·57-s + 0.781·59-s − 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7139584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7139584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7139584\)    =    \(2^{8} \cdot 167^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2672} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7139584,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
167$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 9 T + 43 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T + 31 T^{2} - p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - T + 43 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 61 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 8 T + 77 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 85 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 94 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 21 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T + 83 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 115 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 17 T + 189 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 20 T + 245 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 157 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 9 T + 133 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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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

−8.457337631635048081208002565346, −8.382725397743591804389177484016, −7.958114423324402655419227903907, −7.61421097594578418356217522499, −7.13568508343499727524962763224, −6.69209985320981824150921740501, −6.28536557366232249434706804624, −6.22045255710074564239205228761, −5.57887927496984136866493809621, −5.22514132203385272999856377627, −4.31283093638949506405694297817, −4.30113307120570612629483582965, −3.89486290341990691446118793637, −3.45989368881514867852212270543, −2.98178009582694149725519491649, −2.87154547946237865816102667459, −1.55807522310230759260450286225, −0.978043654229384867749334486503, 0, 0, 0.978043654229384867749334486503, 1.55807522310230759260450286225, 2.87154547946237865816102667459, 2.98178009582694149725519491649, 3.45989368881514867852212270543, 3.89486290341990691446118793637, 4.30113307120570612629483582965, 4.31283093638949506405694297817, 5.22514132203385272999856377627, 5.57887927496984136866493809621, 6.22045255710074564239205228761, 6.28536557366232249434706804624, 6.69209985320981824150921740501, 7.13568508343499727524962763224, 7.61421097594578418356217522499, 7.958114423324402655419227903907, 8.382725397743591804389177484016, 8.457337631635048081208002565346

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