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Properties

Label 4-672e2-1.1-c1e2-0-22
Degree $4$
Conductor $451584$
Sign $1$
Analytic cond. $28.7933$
Root an. cond. $2.31645$
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 − 4·19-s + 2·25-s − 4·27-s + 8·31-s + 4·37-s + 4·43-s + 49-s − 8·57-s − 8·61-s + 28·67-s − 4·73-s + 4·75-s + 8·79-s − 11·81-s + 16·93-s + 20·97-s + 8·103-s + 12·109-s + 8·111-s − 2·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s − 0.917·19-s + 2/5·25-s − 0.769·27-s + 1.43·31-s + 0.657·37-s + 0.609·43-s + 1/7·49-s − 1.05·57-s − 1.02·61-s + 3.42·67-s − 0.468·73-s + 0.461·75-s + 0.900·79-s − 1.22·81-s + 1.65·93-s + 2.03·97-s + 0.788·103-s + 1.14·109-s + 0.759·111-s − 0.181·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(451584\)    =    \(2^{10} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(28.7933\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 451584,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.700585138\)
\(L(\frac12)\) \(\approx\) \(2.700585138\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.11.a_c
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.e_ba
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.29.a_aba
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.31.ai_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.41.a_ao
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.ae_di
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.47.a_o
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.53.a_acw
59$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.59.a_by
61$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.i_es
67$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) 2.67.abc_mo
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.a_bq
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.e_fe
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.79.ai_gc
83$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.83.a_bi
89$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.89.a_c
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.97.au_ks
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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

−8.490774623548777829279143931272, −8.236929819141841651765583148722, −7.76184574695327516296942252395, −7.39662466347816178946335348937, −6.70219109517965929169425331842, −6.36837472820407922258609562378, −5.87251555095505554762785577531, −5.19965403964428379997838793216, −4.64390893417890606282636950921, −4.15500671660247759192407437414, −3.58419662545083671849840604277, −3.02966923238249300624363459220, −2.42215575681526220167540111009, −1.98384498746613515564254023172, −0.835341901598669086898097970815, 0.835341901598669086898097970815, 1.98384498746613515564254023172, 2.42215575681526220167540111009, 3.02966923238249300624363459220, 3.58419662545083671849840604277, 4.15500671660247759192407437414, 4.64390893417890606282636950921, 5.19965403964428379997838793216, 5.87251555095505554762785577531, 6.36837472820407922258609562378, 6.70219109517965929169425331842, 7.39662466347816178946335348937, 7.76184574695327516296942252395, 8.236929819141841651765583148722, 8.490774623548777829279143931272

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