Properties

Label 4-1344e2-1.1-c3e2-0-11
Degree $4$
Conductor $1806336$
Sign $1$
Analytic cond. $6288.26$
Root an. cond. $8.90497$
Motivic weight $3$
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  − 6·3-s − 14·5-s + 14·7-s + 27·9-s − 18·11-s − 48·13-s + 84·15-s + 34·17-s + 16·19-s − 84·21-s + 110·23-s + 74·25-s − 108·27-s − 212·29-s − 136·31-s + 108·33-s − 196·35-s + 24·37-s + 288·39-s + 694·41-s + 584·43-s − 378·45-s − 316·47-s + 147·49-s − 204·51-s − 560·53-s + 252·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.25·5-s + 0.755·7-s + 9-s − 0.493·11-s − 1.02·13-s + 1.44·15-s + 0.485·17-s + 0.193·19-s − 0.872·21-s + 0.997·23-s + 0.591·25-s − 0.769·27-s − 1.35·29-s − 0.787·31-s + 0.569·33-s − 0.946·35-s + 0.106·37-s + 1.18·39-s + 2.64·41-s + 2.07·43-s − 1.25·45-s − 0.980·47-s + 3/7·49-s − 0.560·51-s − 1.45·53-s + 0.617·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(6288.26\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1344} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1806336,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
7$C_1$ \( ( 1 - p T )^{2} \)
good5$D_{4}$ \( 1 + 14 T + 122 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 18 T + 1150 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 48 T + 4262 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 2 p T - 4222 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 16 T + 10950 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 110 T + 18686 T^{2} - 110 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 212 T + 57182 T^{2} + 212 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 136 T + 61374 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 24 T - 18202 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 694 T + 243914 T^{2} - 694 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 584 T + 232950 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 316 T + 231902 T^{2} + 316 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 560 T + 369782 T^{2} + 560 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 492 T + 351622 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 604 T + 406398 T^{2} - 604 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 1020 T + 843926 T^{2} - 1020 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1710 T + 1336222 T^{2} + 1710 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 1312 T + 1201998 T^{2} + 1312 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 556 T + 751134 T^{2} + 556 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 264 T + 979750 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 70 T - 186262 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 136 T + 1812270 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.898388983103280563532982969003, −8.840960751031803822892154813561, −7.83948456397982464219365481451, −7.75314975862125950084012002168, −7.38221321660641608555739934424, −7.32790981849486330907346223526, −6.58572205966979815593786534839, −6.03831266585576194335398888208, −5.58935902445925575368396872861, −5.27507144677204813777710233267, −4.81755376541141603813152416760, −4.44228088863160655126473648652, −3.91109382706976883806622938279, −3.61237687761207064539602668607, −2.63765087168340060138996362987, −2.42319536327476058552345856761, −1.34352988145364680011807430776, −1.01237081630015100837115822755, 0, 0, 1.01237081630015100837115822755, 1.34352988145364680011807430776, 2.42319536327476058552345856761, 2.63765087168340060138996362987, 3.61237687761207064539602668607, 3.91109382706976883806622938279, 4.44228088863160655126473648652, 4.81755376541141603813152416760, 5.27507144677204813777710233267, 5.58935902445925575368396872861, 6.03831266585576194335398888208, 6.58572205966979815593786534839, 7.32790981849486330907346223526, 7.38221321660641608555739934424, 7.75314975862125950084012002168, 7.83948456397982464219365481451, 8.840960751031803822892154813561, 8.898388983103280563532982969003

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