Properties

Label 4-12e6-1.1-c1e2-0-3
Degree $4$
Conductor $2985984$
Sign $1$
Analytic cond. $190.388$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·11-s − 8·13-s + 12·23-s + 3·25-s − 20·37-s + 20·47-s − 49-s − 8·59-s − 16·71-s − 6·73-s + 18·83-s + 14·97-s + 34·107-s − 4·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s − 16·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯
L(s)  = 1  + 0.603·11-s − 2.21·13-s + 2.50·23-s + 3/5·25-s − 3.28·37-s + 2.91·47-s − 1/7·49-s − 1.04·59-s − 1.89·71-s − 0.702·73-s + 1.97·83-s + 1.42·97-s + 3.28·107-s − 0.383·109-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.33·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2985984\)    =    \(2^{12} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(190.388\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2985984,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.747821713\)
\(L(\frac12)\) \(\approx\) \(1.747821713\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 93 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−9.282422527736888303633610128280, −9.121121629782389058986131123389, −8.811300139257596599331322741657, −8.618081729078947053173652392082, −7.66326356957408072686519371675, −7.49859384873538086729347000623, −7.15116899358056774879144304414, −6.89208770407464967596666447518, −6.45140615756740678946649068375, −5.82300039922439497231201252660, −5.29507536267221884638499905672, −5.02545959388557553075024709866, −4.68386016038351739450605908702, −4.22502165299621719669060191330, −3.42673669077228629319182351147, −3.16936477799108320046369173323, −2.57465764992036494543346087747, −2.06429854401777260677487321608, −1.34267636402143208219632230498, −0.51788370474579765160855112023, 0.51788370474579765160855112023, 1.34267636402143208219632230498, 2.06429854401777260677487321608, 2.57465764992036494543346087747, 3.16936477799108320046369173323, 3.42673669077228629319182351147, 4.22502165299621719669060191330, 4.68386016038351739450605908702, 5.02545959388557553075024709866, 5.29507536267221884638499905672, 5.82300039922439497231201252660, 6.45140615756740678946649068375, 6.89208770407464967596666447518, 7.15116899358056774879144304414, 7.49859384873538086729347000623, 7.66326356957408072686519371675, 8.618081729078947053173652392082, 8.811300139257596599331322741657, 9.121121629782389058986131123389, 9.282422527736888303633610128280

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