Properties

Label 4-4400e2-1.1-c1e2-0-32
Degree $4$
Conductor $19360000$
Sign $1$
Analytic cond. $1234.41$
Root an. cond. $5.92740$
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  + 3·3-s + 7-s + 2·9-s − 2·11-s − 8·13-s − 17-s + 3·21-s + 3·23-s − 6·27-s − 5·29-s + 6·31-s − 6·33-s − 16·37-s − 24·39-s − 6·41-s − 12·43-s + 6·47-s − 2·49-s − 3·51-s − 3·53-s − 10·59-s − 11·61-s + 2·63-s + 16·67-s + 9·69-s + 6·71-s − 23·73-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.377·7-s + 2/3·9-s − 0.603·11-s − 2.21·13-s − 0.242·17-s + 0.654·21-s + 0.625·23-s − 1.15·27-s − 0.928·29-s + 1.07·31-s − 1.04·33-s − 2.63·37-s − 3.84·39-s − 0.937·41-s − 1.82·43-s + 0.875·47-s − 2/7·49-s − 0.420·51-s − 0.412·53-s − 1.30·59-s − 1.40·61-s + 0.251·63-s + 1.95·67-s + 1.08·69-s + 0.712·71-s − 2.69·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(19360000\)    =    \(2^{8} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1234.41\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 19360000,\ (\ :1/2, 1/2),\ 1)\)

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$D_{4}$ \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 16 T + 133 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 10 T + 123 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 11 T + 121 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_4$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 23 T + 277 T^{2} + 23 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 5 T + 153 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 27 T + 337 T^{2} + 27 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 25 T + 303 T^{2} + 25 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + T + 193 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−8.259741115214459591493335730899, −7.968275165552492968490092800514, −7.35428668053940891074640686375, −7.34332128446837249426664486604, −6.90522295290704701884676870667, −6.62635956897337541844045736723, −5.74835161620224278970938375701, −5.63953997479103078442159367514, −4.96333895446597968853112499311, −4.95004495714196614097970491129, −4.41632042810119722171234999991, −3.93777399207267259103718415756, −3.25357155132373839925404481607, −3.11576859801955217031266819624, −2.73017203621577420589298247557, −2.37413644325933034098614993449, −1.72256025701593594240909040968, −1.58162896782745415033246615935, 0, 0, 1.58162896782745415033246615935, 1.72256025701593594240909040968, 2.37413644325933034098614993449, 2.73017203621577420589298247557, 3.11576859801955217031266819624, 3.25357155132373839925404481607, 3.93777399207267259103718415756, 4.41632042810119722171234999991, 4.95004495714196614097970491129, 4.96333895446597968853112499311, 5.63953997479103078442159367514, 5.74835161620224278970938375701, 6.62635956897337541844045736723, 6.90522295290704701884676870667, 7.34332128446837249426664486604, 7.35428668053940891074640686375, 7.968275165552492968490092800514, 8.259741115214459591493335730899

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