Properties

Label 4-12e6-1.1-c3e2-0-8
Degree $4$
Conductor $2985984$
Sign $1$
Analytic cond. $10394.8$
Root an. cond. $10.0972$
Motivic weight $3$
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  − 4·5-s − 6·7-s − 52·11-s − 26·13-s + 188·17-s + 74·19-s + 148·23-s − 58·25-s − 288·29-s − 248·31-s + 24·35-s − 342·37-s + 256·43-s + 132·47-s + 61·49-s − 952·53-s + 208·55-s + 1.00e3·59-s + 34·61-s + 104·65-s + 866·67-s + 776·71-s + 1.87e3·73-s + 312·77-s + 182·79-s + 1.33e3·83-s − 752·85-s + ⋯
L(s)  = 1  − 0.357·5-s − 0.323·7-s − 1.42·11-s − 0.554·13-s + 2.68·17-s + 0.893·19-s + 1.34·23-s − 0.463·25-s − 1.84·29-s − 1.43·31-s + 0.115·35-s − 1.51·37-s + 0.907·43-s + 0.409·47-s + 0.177·49-s − 2.46·53-s + 0.509·55-s + 2.21·59-s + 0.0713·61-s + 0.198·65-s + 1.57·67-s + 1.29·71-s + 3.00·73-s + 0.461·77-s + 0.259·79-s + 1.76·83-s − 0.959·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2985984\)    =    \(2^{12} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(10394.8\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2985984,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.642545748\)
\(L(\frac12)\) \(\approx\) \(2.642545748\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 6 T - 25 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 52 T + 1718 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 2 p T + 3843 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 188 T + 18482 T^{2} - 188 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 74 T + 3567 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 148 T + 25310 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 288 T + 57994 T^{2} + 288 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 8 p T + 63438 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 342 T + 112547 T^{2} + 342 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2^2$ \( 1 - 46478 T^{2} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 256 T + 172518 T^{2} - 256 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 132 T + 211822 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 952 T + 489050 T^{2} + 952 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 1004 T + 583382 T^{2} - 1004 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 34 T + 246171 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 866 T + 742935 T^{2} - 866 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 776 T + 704366 T^{2} - 776 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 1874 T + 1630083 T^{2} - 1874 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 182 T + 872679 T^{2} - 182 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1336 T + 1208918 T^{2} - 1336 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 876 T + 1522402 T^{2} - 876 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 38 T + 431787 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.358765312780278401479212376765, −8.824175350331989322194491701653, −8.048091139939526659943047846973, −7.962329316961856912054696509992, −7.59964695747235742277194189476, −7.38699878302609553741124934831, −6.84696376979825856880117968712, −6.45415084552120555923479882554, −5.64389408665174639761484021487, −5.35632669414955620631450574763, −5.21729080026802414317059174454, −5.00070823066020826338269587984, −3.75593557045953240798197042368, −3.75163797201647766083470276473, −3.31639389024373499978039220770, −2.79628043896140670554319807611, −2.16782639712866903561629771123, −1.65376343970714549612173030762, −0.75841012413738186725470397206, −0.48413003409617672723873716607, 0.48413003409617672723873716607, 0.75841012413738186725470397206, 1.65376343970714549612173030762, 2.16782639712866903561629771123, 2.79628043896140670554319807611, 3.31639389024373499978039220770, 3.75163797201647766083470276473, 3.75593557045953240798197042368, 5.00070823066020826338269587984, 5.21729080026802414317059174454, 5.35632669414955620631450574763, 5.64389408665174639761484021487, 6.45415084552120555923479882554, 6.84696376979825856880117968712, 7.38699878302609553741124934831, 7.59964695747235742277194189476, 7.962329316961856912054696509992, 8.048091139939526659943047846973, 8.824175350331989322194491701653, 9.358765312780278401479212376765

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