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Properties

Label 4-528e2-1.1-c1e2-0-26
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 + 4·5-s + 3·9-s + 4·11-s − 8·15-s + 6·25-s − 4·27-s − 8·31-s − 8·33-s − 4·37-s + 12·45-s + 16·47-s − 2·49-s + 4·53-s + 16·55-s + 8·59-s − 12·75-s + 5·81-s + 4·89-s + 16·93-s − 4·97-s + 12·99-s + 8·103-s + 8·111-s + 4·113-s + 5·121-s + 4·125-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 9-s + 1.20·11-s − 2.06·15-s + 6/5·25-s − 0.769·27-s − 1.43·31-s − 1.39·33-s − 0.657·37-s + 1.78·45-s + 2.33·47-s − 2/7·49-s + 0.549·53-s + 2.15·55-s + 1.04·59-s − 1.38·75-s + 5/9·81-s + 0.423·89-s + 1.65·93-s − 0.406·97-s + 1.20·99-s + 0.788·103-s + 0.759·111-s + 0.376·113-s + 5/11·121-s + 0.357·125-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

\(L(1)\) \(\approx\) \(1.842267908\)
\(L(\frac12)\) \(\approx\) \(1.842267908\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.5.ae_k
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.e_ck
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.41.a_aby
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.43.a_bm
47$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.47.aq_fy
53$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.ae_cw
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.ai_eo
61$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.61.a_acs
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.71.a_fi
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.79.a_s
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.a_adm
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.ae_acw
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−9.033671088653985543672121045963, −8.673928873270002702805583593839, −7.83649597707996346193905721918, −7.16883328024884521305817927897, −6.90252504783191013112394010982, −6.38267291016525401060201807960, −5.94114974489728302249172684632, −5.47123562838335717118642622783, −5.37549134847881642057018675984, −4.45309928875018615001905510413, −4.01249239657082882149778823149, −3.26465768207581254079658448645, −2.18309949700366557866941981806, −1.79745069979362293085322200423, −0.912687133475634625402746573576, 0.912687133475634625402746573576, 1.79745069979362293085322200423, 2.18309949700366557866941981806, 3.26465768207581254079658448645, 4.01249239657082882149778823149, 4.45309928875018615001905510413, 5.37549134847881642057018675984, 5.47123562838335717118642622783, 5.94114974489728302249172684632, 6.38267291016525401060201807960, 6.90252504783191013112394010982, 7.16883328024884521305817927897, 7.83649597707996346193905721918, 8.673928873270002702805583593839, 9.033671088653985543672121045963

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