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Properties

Label 4-528e2-1.1-c1e2-0-31
Degree $4$
Conductor $278784$
Sign $1$
Analytic cond. $17.7755$
Root an. cond. $2.05331$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 9-s − 2·11-s + 6·13-s − 8·23-s + 7·25-s − 4·27-s − 4·33-s + 37-s + 12·39-s − 2·47-s − 5·49-s + 16·59-s + 26·61-s − 16·69-s + 18·71-s + 14·73-s + 14·75-s − 11·81-s − 4·83-s + 17·97-s − 2·99-s + 20·107-s − 15·109-s + 2·111-s + 6·117-s − 7·121-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s − 0.603·11-s + 1.66·13-s − 1.66·23-s + 7/5·25-s − 0.769·27-s − 0.696·33-s + 0.164·37-s + 1.92·39-s − 0.291·47-s − 5/7·49-s + 2.08·59-s + 3.32·61-s − 1.92·69-s + 2.13·71-s + 1.63·73-s + 1.61·75-s − 1.22·81-s − 0.439·83-s + 1.72·97-s − 0.201·99-s + 1.93·107-s − 1.43·109-s + 0.189·111-s + 0.554·117-s − 0.636·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(278784\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(17.7755\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 278784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.688244455\)
\(L(\frac12)\) \(\approx\) \(2.688244455\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.5.a_ah
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.13.ag_bj
17$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \) 2.17.a_al
19$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.19.a_r
23$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.i_cg
29$C_2^2$ \( 1 + 45 T^{2} + p^{2} T^{4} \) 2.29.a_bt
31$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.31.a_at
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.37.ab_cu
41$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.41.a_n
43$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.43.a_s
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.c_bu
53$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \) 2.53.a_bd
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.59.aq_gk
61$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) 2.61.aba_lb
67$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \) 2.67.a_ach
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.71.as_ig
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.73.ao_gx
79$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.79.a_z
83$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.e_cs
89$C_2^2$ \( 1 - 107 T^{2} + p^{2} T^{4} \) 2.89.a_aed
97$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.97.ar_ka
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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

−8.654814621778202530505526987837, −8.507198614153391151300763214520, −7.941323314875383662879423465110, −7.86390316837770091490612499013, −6.92347404905887628151458166188, −6.60439652221507293086855313514, −6.08321529200842559813535643995, −5.38580391552787809108339314726, −5.09315447873694041854794598846, −4.08394529100941897329077073139, −3.75162252295450433717642278479, −3.32821977436952177939239431558, −2.45034247269704930097011128614, −2.08613917754322206912493830633, −0.948348301873294688826112043481, 0.948348301873294688826112043481, 2.08613917754322206912493830633, 2.45034247269704930097011128614, 3.32821977436952177939239431558, 3.75162252295450433717642278479, 4.08394529100941897329077073139, 5.09315447873694041854794598846, 5.38580391552787809108339314726, 6.08321529200842559813535643995, 6.60439652221507293086855313514, 6.92347404905887628151458166188, 7.86390316837770091490612499013, 7.941323314875383662879423465110, 8.507198614153391151300763214520, 8.654814621778202530505526987837

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