Properties

Label 4-2736e2-1.1-c1e2-0-13
Degree $4$
Conductor $7485696$
Sign $1$
Analytic cond. $477.294$
Root an. cond. $4.67408$
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  − 8·19-s − 10·25-s + 8·31-s + 2·49-s − 28·61-s + 32·67-s + 20·73-s − 8·79-s + 40·103-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 1.83·19-s − 2·25-s + 1.43·31-s + 2/7·49-s − 3.58·61-s + 3.90·67-s + 2.34·73-s − 0.900·79-s + 3.94·103-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7485696\)    =    \(2^{8} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(477.294\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7485696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.353172910\)
\(L(\frac12)\) \(\approx\) \(1.353172910\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−9.062011207109445203086505410563, −8.555588453497860152279525854744, −8.137073083253253491128724350536, −8.075510122668780086156720606333, −7.51390004933455115710133968594, −7.20558510921975363480825770449, −6.56529551810215557679265974918, −6.30582384584795796831680319876, −6.09561694328624989070662709822, −5.65973933473907663686264215654, −4.90743534700802837146013371229, −4.82723154636183015778244017959, −4.26399833840743041121512714889, −3.74251953912043632090862970789, −3.58764265288227060549944239865, −2.79380017391401252581057271808, −2.21217786533952524625051869433, −2.05525159075940714512799237288, −1.22884542976667090190598622479, −0.38758943591698438846355927388, 0.38758943591698438846355927388, 1.22884542976667090190598622479, 2.05525159075940714512799237288, 2.21217786533952524625051869433, 2.79380017391401252581057271808, 3.58764265288227060549944239865, 3.74251953912043632090862970789, 4.26399833840743041121512714889, 4.82723154636183015778244017959, 4.90743534700802837146013371229, 5.65973933473907663686264215654, 6.09561694328624989070662709822, 6.30582384584795796831680319876, 6.56529551810215557679265974918, 7.20558510921975363480825770449, 7.51390004933455115710133968594, 8.075510122668780086156720606333, 8.137073083253253491128724350536, 8.555588453497860152279525854744, 9.062011207109445203086505410563

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