Properties

Label 4-666e2-1.1-c2e2-0-2
Degree $4$
Conductor $443556$
Sign $1$
Analytic cond. $329.320$
Root an. cond. $4.25995$
Motivic weight $2$
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  − 2·2-s + 2·4-s + 12·5-s + 10·7-s − 24·10-s + 24·13-s − 20·14-s − 4·16-s + 26·17-s + 2·19-s + 24·20-s − 34·23-s + 72·25-s − 48·26-s + 20·28-s + 38·29-s + 72·31-s + 8·32-s − 52·34-s + 120·35-s − 4·38-s − 96·43-s + 68·46-s − 110·47-s − 23·49-s − 144·50-s + 48·52-s + ⋯
L(s)  = 1  − 2-s + 1/2·4-s + 12/5·5-s + 10/7·7-s − 2.39·10-s + 1.84·13-s − 1.42·14-s − 1/4·16-s + 1.52·17-s + 2/19·19-s + 6/5·20-s − 1.47·23-s + 2.87·25-s − 1.84·26-s + 5/7·28-s + 1.31·29-s + 2.32·31-s + 1/4·32-s − 1.52·34-s + 24/7·35-s − 0.105·38-s − 2.23·43-s + 1.47·46-s − 2.34·47-s − 0.469·49-s − 2.87·50-s + 0.923·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(443556\)    =    \(2^{2} \cdot 3^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(329.320\)
Root analytic conductor: \(4.25995\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 443556,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.087961212\)
\(L(\frac12)\) \(\approx\) \(4.087961212\)
\(L(2)\) 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$C_2$ \( 1 + p T + p T^{2} \)
3 \( 1 \)
37$C_2$ \( 1 + p^{2} T^{2} \)
good5$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 217 T^{2} + p^{4} T^{4} \)
13$C_2^2$ \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2^2$ \( 1 - 26 T + 338 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} \)
19$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \)
23$C_2^2$ \( 1 + 34 T + 578 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 38 T + 722 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 72 T + 2592 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 2737 T^{2} + p^{4} T^{4} \)
43$C_2^2$ \( 1 + 96 T + 4608 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2$ \( ( 1 + 55 T + p^{2} T^{2} )^{2} \)
53$C_2$ \( ( 1 - 35 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 108 T + 5832 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 - 1922 T^{2} + p^{4} T^{4} \)
71$C_2$ \( ( 1 + 7 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 1223 T^{2} + p^{4} T^{4} \)
79$C_2^2$ \( 1 - 82 T + 3362 T^{2} - 82 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2$ \( ( 1 - 55 T + p^{2} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 38 T + 722 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} \)
97$C_2^2$ \( 1 + 156 T + 12168 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} \)
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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

−10.20796623488251245303794929422, −9.945372523859589090657878745514, −9.920695510151271500033089245569, −9.416705984551022634628396360356, −8.627085367227942103361385475102, −8.403686719617431735107872372781, −8.046986112715320044895576342385, −7.945487880384032386528758655701, −6.73209242850817660159020748190, −6.47894090518885902944055841425, −6.27851144566100108594430709562, −5.44979587404783888499515649670, −5.35236476005973609958872107252, −4.76166886975588529527637372573, −3.98528762229532403896287749330, −3.20276218120887970433453837370, −2.49883712841250905226300168445, −1.76536679019260333433119479010, −1.43366215152967319625367007109, −1.00610333426589473563970155289, 1.00610333426589473563970155289, 1.43366215152967319625367007109, 1.76536679019260333433119479010, 2.49883712841250905226300168445, 3.20276218120887970433453837370, 3.98528762229532403896287749330, 4.76166886975588529527637372573, 5.35236476005973609958872107252, 5.44979587404783888499515649670, 6.27851144566100108594430709562, 6.47894090518885902944055841425, 6.73209242850817660159020748190, 7.945487880384032386528758655701, 8.046986112715320044895576342385, 8.403686719617431735107872372781, 8.627085367227942103361385475102, 9.416705984551022634628396360356, 9.920695510151271500033089245569, 9.945372523859589090657878745514, 10.20796623488251245303794929422

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