Properties

Label 4-168e2-1.1-c3e2-0-4
Degree $4$
Conductor $28224$
Sign $1$
Analytic cond. $98.2541$
Root an. cond. $3.14838$
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  + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(98.2541\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{168} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28224,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.898199294\)
\(L(\frac12)\) \(\approx\) \(5.898199294\)
\(L(\frac{5}{2})\) 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

−12.82968373319003569046052129900, −12.06888665870260186297538466713, −11.65708535942331082089134661137, −10.94651193254892048556377888356, −10.39378168409657923298955351062, −10.16300705707623329432265438692, −9.275007092314261546216238069534, −9.104576159802684794036175587543, −8.588666547680421905458836578486, −8.109065288083680973331230213696, −7.36519219012300296816301509798, −6.93989463786453034573750374663, −5.90653017967864764418028612321, −5.90615384353472700633124666286, −4.71298521037037433772872947045, −4.32798627258125771807911710005, −3.25237628222443020654370158887, −2.76376987435710596838630071607, −1.64662031906487988199545594164, −1.31445216384837415690922545194, 1.31445216384837415690922545194, 1.64662031906487988199545594164, 2.76376987435710596838630071607, 3.25237628222443020654370158887, 4.32798627258125771807911710005, 4.71298521037037433772872947045, 5.90615384353472700633124666286, 5.90653017967864764418028612321, 6.93989463786453034573750374663, 7.36519219012300296816301509798, 8.109065288083680973331230213696, 8.588666547680421905458836578486, 9.104576159802684794036175587543, 9.275007092314261546216238069534, 10.16300705707623329432265438692, 10.39378168409657923298955351062, 10.94651193254892048556377888356, 11.65708535942331082089134661137, 12.06888665870260186297538466713, 12.82968373319003569046052129900

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