Properties

Label 4-336e2-1.1-c3e2-0-1
Degree $4$
Conductor $112896$
Sign $1$
Analytic cond. $393.016$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 15·5-s − 35·7-s − 9·11-s − 176·13-s − 45·15-s + 84·17-s + 104·19-s + 105·21-s − 84·23-s + 125·25-s + 27·27-s + 102·29-s + 185·31-s + 27·33-s − 525·35-s − 44·37-s + 528·39-s − 336·41-s − 652·43-s − 138·47-s + 882·49-s − 252·51-s − 639·53-s − 135·55-s − 312·57-s + 159·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 1.88·7-s − 0.246·11-s − 3.75·13-s − 0.774·15-s + 1.19·17-s + 1.25·19-s + 1.09·21-s − 0.761·23-s + 25-s + 0.192·27-s + 0.653·29-s + 1.07·31-s + 0.142·33-s − 2.53·35-s − 0.195·37-s + 2.16·39-s − 1.27·41-s − 2.31·43-s − 0.428·47-s + 18/7·49-s − 0.691·51-s − 1.65·53-s − 0.330·55-s − 0.725·57-s + 0.350·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(112896\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(393.016\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 112896,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.07097981497\)
\(L(\frac12)\) \(\approx\) \(0.07097981497\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p T + p^{2} T^{2} \)
7$C_2$ \( 1 + 5 p T + p^{3} T^{2} \)
good5$C_2^2$ \( 1 - 3 p T + 4 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 9 T - 1250 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 88 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 104 T + 3957 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 84 T - 5111 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 51 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 185 T + 4434 T^{2} - 185 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 44 T - 48717 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 168 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 326 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 138 T - 84779 T^{2} + 138 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 639 T + 259444 T^{2} + 639 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 159 T - 180098 T^{2} - 159 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 722 T + 294303 T^{2} + 722 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 166 T - 273207 T^{2} + 166 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 1086 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 218 T - 341493 T^{2} + 218 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 583 T - 153150 T^{2} + 583 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 597 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 1038 T + 372475 T^{2} - 1038 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 169 T + p^{3} T^{2} )^{2} \)
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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

−11.83391316897089071891655080036, −10.39757026642541822150939783551, −10.25576065398391225757010550500, −9.944000680502086649920922572583, −9.720259096452749960556992154629, −9.424134482519009419504758690818, −8.744761085666186099424341067458, −7.71331329562421992596159268103, −7.59281829393495495719420288448, −6.78905655987627756655892213963, −6.63525632926821996004962681336, −5.91382134062253626047663422255, −5.52116745542852503046027327156, −4.86765044910558577603093834372, −4.73722222220035555290693618091, −3.24117886410597422240745604571, −2.99703763955553344261495798388, −2.42983144822964596863136607039, −1.47184416633862354614292300231, −0.094817040078483643258617881656, 0.094817040078483643258617881656, 1.47184416633862354614292300231, 2.42983144822964596863136607039, 2.99703763955553344261495798388, 3.24117886410597422240745604571, 4.73722222220035555290693618091, 4.86765044910558577603093834372, 5.52116745542852503046027327156, 5.91382134062253626047663422255, 6.63525632926821996004962681336, 6.78905655987627756655892213963, 7.59281829393495495719420288448, 7.71331329562421992596159268103, 8.744761085666186099424341067458, 9.424134482519009419504758690818, 9.720259096452749960556992154629, 9.944000680502086649920922572583, 10.25576065398391225757010550500, 10.39757026642541822150939783551, 11.83391316897089071891655080036

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