Properties

Label 4-304e2-1.1-c1e2-0-12
Degree $4$
Conductor $92416$
Sign $1$
Analytic cond. $5.89252$
Root an. cond. $1.55802$
Motivic weight $1$
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-s + 8·7-s + 3·9-s − 6·11-s − 2·13-s + 6·17-s + 7·19-s + 8·21-s − 6·23-s + 5·25-s + 8·27-s − 4·31-s − 6·33-s − 20·37-s − 2·39-s − 9·41-s − 4·43-s + 34·49-s + 6·51-s − 6·53-s + 7·57-s − 9·59-s + 4·61-s + 24·63-s − 7·67-s − 6·69-s − 6·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 3.02·7-s + 9-s − 1.80·11-s − 0.554·13-s + 1.45·17-s + 1.60·19-s + 1.74·21-s − 1.25·23-s + 25-s + 1.53·27-s − 0.718·31-s − 1.04·33-s − 3.28·37-s − 0.320·39-s − 1.40·41-s − 0.609·43-s + 34/7·49-s + 0.840·51-s − 0.824·53-s + 0.927·57-s − 1.17·59-s + 0.512·61-s + 3.02·63-s − 0.855·67-s − 0.722·69-s − 0.712·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(92416\)    =    \(2^{8} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5.89252\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 92416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.562255824\)
\(L(\frac12)\) \(\approx\) \(2.562255824\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 - 7 T + p T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
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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

−11.79046945920849194084307987192, −11.70267975460397978759753019787, −10.73359721735879179389445418935, −10.71847297110403078461802098912, −10.03159301912368317158478020777, −9.932640092240573728227139780063, −8.879975529068642090755983407217, −8.464498472816492501032971920398, −8.064793730236352487603491001056, −7.81492517044600848278144870102, −7.18390465856659408311806978343, −7.12523186046341795950573104520, −5.56460508320998101200922413237, −5.37100781141545302116005837049, −4.80639296637173689723380906051, −4.66394074691694157525008378116, −3.49310160595025062941920708005, −2.90543307958866180738477720057, −1.74890550306580313603717464941, −1.55470481856806376749536800444, 1.55470481856806376749536800444, 1.74890550306580313603717464941, 2.90543307958866180738477720057, 3.49310160595025062941920708005, 4.66394074691694157525008378116, 4.80639296637173689723380906051, 5.37100781141545302116005837049, 5.56460508320998101200922413237, 7.12523186046341795950573104520, 7.18390465856659408311806978343, 7.81492517044600848278144870102, 8.064793730236352487603491001056, 8.464498472816492501032971920398, 8.879975529068642090755983407217, 9.932640092240573728227139780063, 10.03159301912368317158478020777, 10.71847297110403078461802098912, 10.73359721735879179389445418935, 11.70267975460397978759753019787, 11.79046945920849194084307987192

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