Properties

Label 4-12e6-1.1-c3e2-0-1
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

\(L(2)\) \(\approx\) \(0.5088801949\)
\(L(\frac12)\) \(\approx\) \(0.5088801949\)
\(L(\frac{5}{2})\) 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

−9.210981706177144223936553175816, −8.797001251532814768053227947356, −8.146562049973368067210063741889, −8.037734608372937347590407404934, −7.53888082995470534991793904964, −7.45288748471507346510946204860, −6.59264484946358436920412706658, −6.45449177217074427172400006373, −5.86420357352723047894603606616, −5.67554976613937624703579401705, −4.92729727852737634473357027064, −4.71110847689193430532551465012, −4.09013391643110489201823776612, −3.59322894492606852207852726764, −3.42665206386592590894863730564, −2.82290041078521775750871177476, −1.91065763311010522864532296909, −1.59435355266943973355241035071, −1.18119744194327315900281415741, −0.15447576520705160532441018235, 0.15447576520705160532441018235, 1.18119744194327315900281415741, 1.59435355266943973355241035071, 1.91065763311010522864532296909, 2.82290041078521775750871177476, 3.42665206386592590894863730564, 3.59322894492606852207852726764, 4.09013391643110489201823776612, 4.71110847689193430532551465012, 4.92729727852737634473357027064, 5.67554976613937624703579401705, 5.86420357352723047894603606616, 6.45449177217074427172400006373, 6.59264484946358436920412706658, 7.45288748471507346510946204860, 7.53888082995470534991793904964, 8.037734608372937347590407404934, 8.146562049973368067210063741889, 8.797001251532814768053227947356, 9.210981706177144223936553175816

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