Properties

Label 4-960e2-1.1-c2e2-0-3
Degree $4$
Conductor $921600$
Sign $1$
Analytic cond. $684.246$
Root an. cond. $5.11449$
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  − 4·3-s + 4·7-s + 7·9-s − 16·13-s + 68·19-s − 16·21-s − 5·25-s + 8·27-s + 28·31-s − 112·37-s + 64·39-s − 16·43-s − 86·49-s − 272·57-s + 92·61-s + 28·63-s − 64·67-s − 212·73-s + 20·75-s − 44·79-s − 95·81-s − 64·91-s − 112·93-s + 244·97-s − 92·103-s − 172·109-s + 448·111-s + ⋯
L(s)  = 1  − 4/3·3-s + 4/7·7-s + 7/9·9-s − 1.23·13-s + 3.57·19-s − 0.761·21-s − 1/5·25-s + 8/27·27-s + 0.903·31-s − 3.02·37-s + 1.64·39-s − 0.372·43-s − 1.75·49-s − 4.77·57-s + 1.50·61-s + 4/9·63-s − 0.955·67-s − 2.90·73-s + 4/15·75-s − 0.556·79-s − 1.17·81-s − 0.703·91-s − 1.20·93-s + 2.51·97-s − 0.893·103-s − 1.57·109-s + 4.03·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(684.246\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9938868037\)
\(L(\frac12)\) \(\approx\) \(0.9938868037\)
\(L(2)\) 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$C_2$ \( 1 + 4 T + p^{2} T^{2} \)
5$C_2$ \( 1 + p T^{2} \)
good7$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 62 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 398 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 34 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 562 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 62 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 56 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 2642 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2798 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 3998 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 6782 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 46 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 32 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 7202 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 106 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 22 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 802 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \)
97$C_2$ \( ( 1 - 122 T + p^{2} T^{2} )^{2} \)
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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

−10.24782610220659871149923157772, −9.705036937464569071749571404752, −9.456582575263715267585242820110, −8.633970359932460989212575507850, −8.540313096315016142133425429182, −7.60054242649656119659032606813, −7.54612093480690035841000551630, −7.11048432436237405698794120653, −6.68928295667757949413786333596, −6.09602535962335190624310227400, −5.52824387187321171465181983749, −5.23172816797888894396580101117, −4.97228266640970187732595685077, −4.61925192641444519660858165194, −3.73107779751504392107274286673, −3.15944751268388825804045014826, −2.76015719354215676684179659340, −1.66115977713216637502098421771, −1.26814734680581366092497485280, −0.37828556931405939858838502408, 0.37828556931405939858838502408, 1.26814734680581366092497485280, 1.66115977713216637502098421771, 2.76015719354215676684179659340, 3.15944751268388825804045014826, 3.73107779751504392107274286673, 4.61925192641444519660858165194, 4.97228266640970187732595685077, 5.23172816797888894396580101117, 5.52824387187321171465181983749, 6.09602535962335190624310227400, 6.68928295667757949413786333596, 7.11048432436237405698794120653, 7.54612093480690035841000551630, 7.60054242649656119659032606813, 8.540313096315016142133425429182, 8.633970359932460989212575507850, 9.456582575263715267585242820110, 9.705036937464569071749571404752, 10.24782610220659871149923157772

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