Properties

Label 4-4608e2-1.1-c1e2-0-46
Degree $4$
Conductor $21233664$
Sign $1$
Analytic cond. $1353.87$
Root an. cond. $6.06589$
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  + 2·5-s + 8·11-s + 6·13-s + 12·17-s + 8·19-s + 2·25-s + 6·29-s − 8·31-s + 2·37-s + 8·43-s − 16·47-s − 2·49-s − 14·53-s + 16·55-s − 6·61-s + 12·65-s − 16·67-s + 24·79-s − 8·83-s + 24·85-s + 16·95-s + 16·97-s − 14·101-s − 16·107-s + 10·109-s + 32·113-s + 32·121-s + ⋯
L(s)  = 1  + 0.894·5-s + 2.41·11-s + 1.66·13-s + 2.91·17-s + 1.83·19-s + 2/5·25-s + 1.11·29-s − 1.43·31-s + 0.328·37-s + 1.21·43-s − 2.33·47-s − 2/7·49-s − 1.92·53-s + 2.15·55-s − 0.768·61-s + 1.48·65-s − 1.95·67-s + 2.70·79-s − 0.878·83-s + 2.60·85-s + 1.64·95-s + 1.62·97-s − 1.39·101-s − 1.54·107-s + 0.957·109-s + 3.01·113-s + 2.90·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21233664\)    =    \(2^{18} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1353.87\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4608} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21233664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.267287148\)
\(L(\frac12)\) \(\approx\) \(7.267287148\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−8.569028267117341858163542905323, −7.947885281363358547833914804977, −7.84732912415814948894030486778, −7.56036711427846394512089693224, −6.87265703637919469004909322905, −6.67591611004168398158490410375, −6.23393742597915424383474765373, −5.94037993955020654453763340136, −5.71903883616252366042777508005, −5.33720277596916778165013425701, −4.78481354215264691044535901634, −4.45528343281635513792574067847, −3.64492984390797301479360053883, −3.63586968003585771744139013649, −3.08088799824196311276220267706, −3.07601062391924803363290300885, −1.81042524086582731211086656721, −1.58221874533219103996538297112, −1.10069760573670121670120322937, −0.955233362683064482232371904822, 0.955233362683064482232371904822, 1.10069760573670121670120322937, 1.58221874533219103996538297112, 1.81042524086582731211086656721, 3.07601062391924803363290300885, 3.08088799824196311276220267706, 3.63586968003585771744139013649, 3.64492984390797301479360053883, 4.45528343281635513792574067847, 4.78481354215264691044535901634, 5.33720277596916778165013425701, 5.71903883616252366042777508005, 5.94037993955020654453763340136, 6.23393742597915424383474765373, 6.67591611004168398158490410375, 6.87265703637919469004909322905, 7.56036711427846394512089693224, 7.84732912415814948894030486778, 7.947885281363358547833914804977, 8.569028267117341858163542905323

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