Properties

Label 4-48e2-1.1-c1e2-0-2
Degree $4$
Conductor $2304$
Sign $1$
Analytic cond. $0.146905$
Root an. cond. $0.619097$
Motivic weight $1$
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  − 3·9-s − 4·13-s + 10·25-s − 20·37-s + 2·49-s + 28·61-s + 20·73-s + 9·81-s − 28·97-s − 4·109-s + 12·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 9-s − 1.10·13-s + 2·25-s − 3.28·37-s + 2/7·49-s + 3.58·61-s + 2.34·73-s + 81-s − 2.84·97-s − 0.383·109-s + 1.10·117-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 − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(0.146905\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{48} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2304,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6385144647\)
\(L(\frac12)\) \(\approx\) \(0.6385144647\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−15.92369172058273067985032747780, −15.44375138708661754247537486481, −14.60504122510036768778392573017, −14.46956971764483229019315726323, −13.83864131223898957951710912710, −13.12925026034704855482017874281, −12.24056695896816489745402282108, −12.23659147913522161754220310176, −11.26412486568634322020379293105, −10.73331120502494611207295480445, −10.10407073322686381731412723868, −9.338892356501978764549936387737, −8.625402204712350869792689549676, −8.201099013122993350293745119564, −7.02451456717377461766522669630, −6.74418422965018797428766625394, −5.37448383935061293331068662521, −5.06935218801281721891351547462, −3.63537616080026223751227237238, −2.52494733590377164458054694339, 2.52494733590377164458054694339, 3.63537616080026223751227237238, 5.06935218801281721891351547462, 5.37448383935061293331068662521, 6.74418422965018797428766625394, 7.02451456717377461766522669630, 8.201099013122993350293745119564, 8.625402204712350869792689549676, 9.338892356501978764549936387737, 10.10407073322686381731412723868, 10.73331120502494611207295480445, 11.26412486568634322020379293105, 12.23659147913522161754220310176, 12.24056695896816489745402282108, 13.12925026034704855482017874281, 13.83864131223898957951710912710, 14.46956971764483229019315726323, 14.60504122510036768778392573017, 15.44375138708661754247537486481, 15.92369172058273067985032747780

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