Properties

Degree $4$
Conductor $1806336$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

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: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{1344} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1806336,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.581535278\)
\(L(\frac12)\) \(\approx\) \(1.581535278\)
\(L(\frac{5}{2})\) 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

−9.456843256988379242286409583716, −9.103484428239763757380815784192, −8.513283879156760018736000669286, −8.255310701113584102055737365140, −7.70210732346850764069569983535, −7.62864979551680821147675883056, −7.12813488032768843761859641876, −6.81770061740201026603753340311, −6.04101089281835508892110668458, −5.95474117321141810972155710062, −5.12344112340877457819572134439, −4.44924565826895101859961790184, −4.32126864381963202747234487950, −3.77271664871619328424593690309, −3.25600048542501633008371782151, −3.07070072674317224582188115469, −2.26689847740700276137625739031, −1.90649179891262216776572216233, −0.983320536246301504377058832985, −0.30179698712265435674489547231, 0.30179698712265435674489547231, 0.983320536246301504377058832985, 1.90649179891262216776572216233, 2.26689847740700276137625739031, 3.07070072674317224582188115469, 3.25600048542501633008371782151, 3.77271664871619328424593690309, 4.32126864381963202747234487950, 4.44924565826895101859961790184, 5.12344112340877457819572134439, 5.95474117321141810972155710062, 6.04101089281835508892110668458, 6.81770061740201026603753340311, 7.12813488032768843761859641876, 7.62864979551680821147675883056, 7.70210732346850764069569983535, 8.255310701113584102055737365140, 8.513283879156760018736000669286, 9.103484428239763757380815784192, 9.456843256988379242286409583716

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