Properties

Label 4-624e2-1.1-c3e2-0-5
Degree $4$
Conductor $389376$
Sign $1$
Analytic cond. $1355.50$
Root an. cond. $6.06771$
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 − 18·7-s − 90·11-s − 65·13-s + 117·17-s + 42·19-s + 54·21-s + 18·23-s + 223·25-s + 27·27-s + 99·29-s + 270·33-s + 195·37-s + 195·39-s − 63·41-s − 82·43-s − 127·49-s − 351·51-s − 522·53-s − 126·57-s + 1.36e3·59-s + 719·61-s + 1.21e3·67-s − 54·69-s + 810·71-s − 669·75-s + 1.62e3·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.971·7-s − 2.46·11-s − 1.38·13-s + 1.66·17-s + 0.507·19-s + 0.561·21-s + 0.163·23-s + 1.78·25-s + 0.192·27-s + 0.633·29-s + 1.42·33-s + 0.866·37-s + 0.800·39-s − 0.239·41-s − 0.290·43-s − 0.370·49-s − 0.963·51-s − 1.35·53-s − 0.292·57-s + 3.01·59-s + 1.50·61-s + 2.22·67-s − 0.0942·69-s + 1.35·71-s − 1.02·75-s + 2.39·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(389376\)    =    \(2^{8} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1355.50\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 389376,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.511831553\)
\(L(\frac12)\) \(\approx\) \(1.511831553\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p T + p^{2} T^{2} \)
13$C_2$ \( 1 + 5 p T + p^{3} T^{2} \)
good5$C_2^2$ \( 1 - 223 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 + 18 T + 451 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 90 T + 4031 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 117 T + 8776 T^{2} - 117 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 42 T + 7447 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 18 T - 11843 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 - 99 T - 14588 T^{2} - 99 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 21950 T^{2} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 195 T + 63328 T^{2} - 195 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2^2$ \( 1 + 63 T + 70244 T^{2} + 63 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 82 T - 72783 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 202354 T^{2} + p^{6} T^{4} \)
53$C_2$ \( ( 1 + 261 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 1368 T + 829187 T^{2} - 1368 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2$ \( ( 1 - 901 T + p^{3} T^{2} )( 1 + 182 T + p^{3} T^{2} ) \)
67$C_2^2$ \( 1 - 1218 T + 795271 T^{2} - 1218 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2^2$ \( 1 - 810 T + 576611 T^{2} - 810 p^{3} T^{3} + p^{6} T^{4} \)
73$C_2^2$ \( 1 - 309959 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 440 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 284726 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 - 2628 T + 3007097 T^{2} - 2628 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2^2$ \( 1 + 2004 T + 2251345 T^{2} + 2004 p^{3} T^{3} + p^{6} T^{4} \)
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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

−10.34374121828936128909011264142, −9.978054246311598262815684125352, −9.608333577932919263002498478727, −9.545536924990670463170219257098, −8.476492507194088429748778599061, −8.082136654031600065174517999545, −7.942318623864111469084977677969, −7.24658401049696281176947893257, −6.80771346164178141153260604632, −6.56978221886597015938327675212, −5.53266283158403716782740069701, −5.49064688173042492403773877918, −5.02596931636481159153830956279, −4.69712234304784960951449528468, −3.61319423324662773162972473713, −3.14179778991720255905564109631, −2.66209880114558223685616071621, −2.22764543094477287694961305518, −0.790184876162296651817113842163, −0.53185232998327230977231932157, 0.53185232998327230977231932157, 0.790184876162296651817113842163, 2.22764543094477287694961305518, 2.66209880114558223685616071621, 3.14179778991720255905564109631, 3.61319423324662773162972473713, 4.69712234304784960951449528468, 5.02596931636481159153830956279, 5.49064688173042492403773877918, 5.53266283158403716782740069701, 6.56978221886597015938327675212, 6.80771346164178141153260604632, 7.24658401049696281176947893257, 7.942318623864111469084977677969, 8.082136654031600065174517999545, 8.476492507194088429748778599061, 9.545536924990670463170219257098, 9.608333577932919263002498478727, 9.978054246311598262815684125352, 10.34374121828936128909011264142

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