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Properties

Label 4-3024e2-1.1-c1e2-0-18
Degree $4$
Conductor $9144576$
Sign $1$
Analytic cond. $583.066$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 7-s − 3·11-s + 9·13-s − 8·19-s − 9·23-s + 25-s − 9·29-s − 7·31-s − 3·35-s + 20·37-s − 9·41-s + 15·43-s − 3·47-s − 6·49-s + 12·53-s − 9·55-s + 3·59-s − 3·61-s + 27·65-s − 15·67-s + 3·77-s − 9·79-s + 9·83-s − 9·91-s − 24·95-s + 9·97-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.377·7-s − 0.904·11-s + 2.49·13-s − 1.83·19-s − 1.87·23-s + 1/5·25-s − 1.67·29-s − 1.25·31-s − 0.507·35-s + 3.28·37-s − 1.40·41-s + 2.28·43-s − 0.437·47-s − 6/7·49-s + 1.64·53-s − 1.21·55-s + 0.390·59-s − 0.384·61-s + 3.34·65-s − 1.83·67-s + 0.341·77-s − 1.01·79-s + 0.987·83-s − 0.943·91-s − 2.46·95-s + 0.913·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(583.066\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.102436503\)
\(L(\frac12)\) \(\approx\) \(2.102436503\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_i
11$C_2^2$ \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_o
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.aj_bo
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.23.j_by
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.29.j_ca
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.h_s
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2^2$ \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_cq
43$C_2^2$ \( 1 - 15 T + 118 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.43.ap_eo
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.47.d_abm
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.59.ad_aby
61$C_2^2$ \( 1 + 3 T + 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.61.d_cm
67$C_2^2$ \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.67.p_fm
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.71.a_afa
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.73.a_adu
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.j_ec
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.89.a_afa
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.aj_eu
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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

−8.908232751784930249004925663313, −8.527372643489556989156611687472, −8.275035971563430182137155196151, −7.67382767602000676938870667172, −7.66865494263718586508276394785, −6.94303417647998704289492004826, −6.32114591007521894302867344920, −6.18860462804275578083354241005, −5.97047881742530855964557493459, −5.60104207859355831702712068392, −5.40821641362998548598134488918, −4.42829677907149989989878773735, −4.15340819026377159399983559325, −3.90796459580385893500329199951, −3.35650094412952630935165014489, −2.70081615545845661844132755310, −2.22419312307647980801377462830, −1.86125593878462634305661982896, −1.40927311305932208209600465155, −0.43611719011865579707249644283, 0.43611719011865579707249644283, 1.40927311305932208209600465155, 1.86125593878462634305661982896, 2.22419312307647980801377462830, 2.70081615545845661844132755310, 3.35650094412952630935165014489, 3.90796459580385893500329199951, 4.15340819026377159399983559325, 4.42829677907149989989878773735, 5.40821641362998548598134488918, 5.60104207859355831702712068392, 5.97047881742530855964557493459, 6.18860462804275578083354241005, 6.32114591007521894302867344920, 6.94303417647998704289492004826, 7.66865494263718586508276394785, 7.67382767602000676938870667172, 8.275035971563430182137155196151, 8.527372643489556989156611687472, 8.908232751784930249004925663313

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