Properties

Label 4-114e2-1.1-c1e2-0-9
Degree $4$
Conductor $12996$
Sign $1$
Analytic cond. $0.828636$
Root an. cond. $0.954093$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 6·5-s − 4·7-s − 8-s − 3·9-s + 6·10-s − 6·13-s − 4·14-s − 16-s + 12·17-s − 3·18-s + 19-s + 19·25-s − 6·26-s − 6·29-s + 12·34-s − 24·35-s + 38-s − 6·40-s − 3·41-s − 8·43-s − 18·45-s − 6·47-s − 2·49-s + 19·50-s + 6·53-s + 4·56-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.68·5-s − 1.51·7-s − 0.353·8-s − 9-s + 1.89·10-s − 1.66·13-s − 1.06·14-s − 1/4·16-s + 2.91·17-s − 0.707·18-s + 0.229·19-s + 19/5·25-s − 1.17·26-s − 1.11·29-s + 2.05·34-s − 4.05·35-s + 0.162·38-s − 0.948·40-s − 0.468·41-s − 1.21·43-s − 2.68·45-s − 0.875·47-s − 2/7·49-s + 2.68·50-s + 0.824·53-s + 0.534·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12996\)    =    \(2^{2} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.828636\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{114} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.673055844\)
\(L(\frac12)\) \(\approx\) \(1.673055844\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + p T^{2} \)
19$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 9 T + 94 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 5 T - 48 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 24 T + 271 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 139 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
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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

−13.92835200642126841447546786297, −13.26631228244715344175451384191, −12.98303247760488020421352443880, −12.55024267937959604068984170372, −12.02214196569113067546512546877, −11.44732127937116549735938712841, −10.16640429572740401937683306962, −10.15343836154507203378993261647, −9.649821626580960755746413617885, −9.491621912540238649998979429082, −8.708621184955583284032130744406, −7.75230985102298324916190910150, −6.97894429427254294578367153825, −6.28164355707810921589486350585, −5.76995324994470245826656273727, −5.47662583494971535382675742140, −4.98221653212859648678770452758, −3.24426703404144333301740499515, −3.05631423872479306912342985543, −1.95196628564662745654120187475, 1.95196628564662745654120187475, 3.05631423872479306912342985543, 3.24426703404144333301740499515, 4.98221653212859648678770452758, 5.47662583494971535382675742140, 5.76995324994470245826656273727, 6.28164355707810921589486350585, 6.97894429427254294578367153825, 7.75230985102298324916190910150, 8.708621184955583284032130744406, 9.491621912540238649998979429082, 9.649821626580960755746413617885, 10.15343836154507203378993261647, 10.16640429572740401937683306962, 11.44732127937116549735938712841, 12.02214196569113067546512546877, 12.55024267937959604068984170372, 12.98303247760488020421352443880, 13.26631228244715344175451384191, 13.92835200642126841447546786297

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