Properties

Label 4-288e2-1.1-c2e2-0-4
Degree $4$
Conductor $82944$
Sign $1$
Analytic cond. $61.5821$
Root an. cond. $2.80132$
Motivic weight $2$
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  + 48·13-s + 48·25-s + 140·37-s − 98·49-s − 44·61-s − 192·73-s − 288·97-s − 240·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.39e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 3.69·13-s + 1.91·25-s + 3.78·37-s − 2·49-s − 0.721·61-s − 2.63·73-s − 2.96·97-s − 2.20·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 8.22·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(82944\)    =    \(2^{10} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(61.5821\)
Root analytic conductor: \(2.80132\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 82944,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.862792754\)
\(L(\frac12)\) \(\approx\) \(2.862792754\)
\(L(2)\) 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 \( 1 \)
good5$C_2^2$ \( 1 - 48 T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
13$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 480 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
29$C_2^2$ \( 1 - 1680 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 70 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 1440 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_2^2$ \( 1 + 5040 T^{2} + p^{4} T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
61$C_2$ \( ( 1 + 22 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + 96 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
89$C_2^2$ \( 1 + 12480 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 + 144 T + p^{2} T^{2} )^{2} \)
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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

−11.42697745112321182878120258018, −11.40455288084448529170697657602, −10.86894351261976665100392060064, −10.73444681161266803411605178172, −9.905915598574959002864075178796, −9.430786796458750080347435256997, −8.761097516980137156698371596482, −8.647113939972459696625117725755, −8.034030674122440330313574295014, −7.64792360165116751020659223077, −6.53885104430615464515509703611, −6.53352508491681543213215877957, −5.92560697305733138129199407580, −5.44718075636809267901643654807, −4.32269421092743956461936916476, −4.24301766966859931064342635227, −3.23990056100399394213424254890, −2.88628165958439073310052056447, −1.44704203259757710934521997889, −1.00765691716622985797871660799, 1.00765691716622985797871660799, 1.44704203259757710934521997889, 2.88628165958439073310052056447, 3.23990056100399394213424254890, 4.24301766966859931064342635227, 4.32269421092743956461936916476, 5.44718075636809267901643654807, 5.92560697305733138129199407580, 6.53352508491681543213215877957, 6.53885104430615464515509703611, 7.64792360165116751020659223077, 8.034030674122440330313574295014, 8.647113939972459696625117725755, 8.761097516980137156698371596482, 9.430786796458750080347435256997, 9.905915598574959002864075178796, 10.73444681161266803411605178172, 10.86894351261976665100392060064, 11.40455288084448529170697657602, 11.42697745112321182878120258018

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