Properties

Label 4-5376e2-1.1-c1e2-0-19
Degree $4$
Conductor $28901376$
Sign $1$
Analytic cond. $1842.77$
Root an. cond. $6.55191$
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  + 2·7-s − 9-s − 4·17-s − 8·23-s + 6·25-s + 16·31-s + 4·41-s − 16·47-s + 3·49-s − 2·63-s − 24·71-s + 28·73-s + 16·79-s + 81-s + 4·89-s + 20·97-s + 36·113-s − 8·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + ⋯
L(s)  = 1  + 0.755·7-s − 1/3·9-s − 0.970·17-s − 1.66·23-s + 6/5·25-s + 2.87·31-s + 0.624·41-s − 2.33·47-s + 3/7·49-s − 0.251·63-s − 2.84·71-s + 3.27·73-s + 1.80·79-s + 1/9·81-s + 0.423·89-s + 2.03·97-s + 3.38·113-s − 0.733·119-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 + 0.323·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28901376\)    =    \(2^{16} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1842.77\)
Root analytic conductor: \(6.55191\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28901376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.871513647\)
\(L(\frac12)\) \(\approx\) \(2.871513647\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−8.215735335379379103671220428565, −8.158964393260826818419465510316, −7.64228857986017205027178762763, −7.49143038830493183463750356596, −6.74931624508696532361323359352, −6.51308361275404631690947883969, −6.31271659530698073193551938332, −5.96007335712832069188690337281, −5.44025276299990909053513031805, −4.84631101945086933743168470072, −4.74899411191433050641089228391, −4.50976349643955564460695762722, −3.93585234750388257822869608901, −3.47877529801730529661271223687, −3.00009693610116623574686008236, −2.60072437451123026833339294951, −2.03712889503776471748621867268, −1.80898102363758655676791420763, −0.966433392124085248447454948145, −0.51952603050619733549010079545, 0.51952603050619733549010079545, 0.966433392124085248447454948145, 1.80898102363758655676791420763, 2.03712889503776471748621867268, 2.60072437451123026833339294951, 3.00009693610116623574686008236, 3.47877529801730529661271223687, 3.93585234750388257822869608901, 4.50976349643955564460695762722, 4.74899411191433050641089228391, 4.84631101945086933743168470072, 5.44025276299990909053513031805, 5.96007335712832069188690337281, 6.31271659530698073193551938332, 6.51308361275404631690947883969, 6.74931624508696532361323359352, 7.49143038830493183463750356596, 7.64228857986017205027178762763, 8.158964393260826818419465510316, 8.215735335379379103671220428565

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