Properties

Label 4-6240e2-1.1-c1e2-0-15
Degree $4$
Conductor $38937600$
Sign $1$
Analytic cond. $2482.69$
Root an. cond. $7.05879$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s − 3·7-s + 3·9-s + 3·11-s − 2·13-s − 4·15-s − 5·17-s − 4·19-s − 6·21-s + 23-s + 3·25-s + 4·27-s + 6·29-s + 2·31-s + 6·33-s + 6·35-s + 3·37-s − 4·39-s + 5·41-s − 2·43-s − 6·45-s − 6·47-s + 49-s − 10·51-s − 21·53-s − 6·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 1.13·7-s + 9-s + 0.904·11-s − 0.554·13-s − 1.03·15-s − 1.21·17-s − 0.917·19-s − 1.30·21-s + 0.208·23-s + 3/5·25-s + 0.769·27-s + 1.11·29-s + 0.359·31-s + 1.04·33-s + 1.01·35-s + 0.493·37-s − 0.640·39-s + 0.780·41-s − 0.304·43-s − 0.894·45-s − 0.875·47-s + 1/7·49-s − 1.40·51-s − 2.88·53-s − 0.809·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(38937600\)    =    \(2^{10} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2482.69\)
Root analytic conductor: \(7.05879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 38937600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - T + 38 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 3 T + 68 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 5 T + 80 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 21 T + 208 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 14 T + 150 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 136 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 25 T + 306 T^{2} + 25 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 18 T + 214 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 9 T + 124 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - T + 120 T^{2} - p T^{3} + p^{2} T^{4} \)
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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

−7.944402474334000709144354697254, −7.65850080870521988519817595890, −6.94309427515614049687156616942, −6.87614989520827523125175693594, −6.63782722916762836552082879154, −6.38843838440150210586119062886, −5.81532768465000277304388660323, −5.34780925261638376154217830821, −4.71289872935930689991358443695, −4.44439984543876444517338429023, −4.10707421330271105895252393809, −3.98674035312219027409459734045, −3.20792184020664322468012143805, −3.10888672306195319454915802463, −2.56182931708867292820606685710, −2.40479157068419456330073465136, −1.44043133065990194696224928943, −1.28646576775690583960011701612, 0, 0, 1.28646576775690583960011701612, 1.44043133065990194696224928943, 2.40479157068419456330073465136, 2.56182931708867292820606685710, 3.10888672306195319454915802463, 3.20792184020664322468012143805, 3.98674035312219027409459734045, 4.10707421330271105895252393809, 4.44439984543876444517338429023, 4.71289872935930689991358443695, 5.34780925261638376154217830821, 5.81532768465000277304388660323, 6.38843838440150210586119062886, 6.63782722916762836552082879154, 6.87614989520827523125175693594, 6.94309427515614049687156616942, 7.65850080870521988519817595890, 7.944402474334000709144354697254

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