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Properties

Label 4-24e4-1.1-c1e2-0-15
Degree $4$
Conductor $331776$
Sign $1$
Analytic cond. $21.1543$
Root an. cond. $2.14461$
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  + 4·7-s − 4·13-s − 8·19-s − 2·25-s + 4·31-s + 12·37-s + 8·43-s + 2·49-s + 12·61-s + 12·73-s + 4·79-s − 16·91-s − 20·97-s + 12·103-s + 12·109-s + 10·121-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯
L(s)  = 1  + 1.51·7-s − 1.10·13-s − 1.83·19-s − 2/5·25-s + 0.718·31-s + 1.97·37-s + 1.21·43-s + 2/7·49-s + 1.53·61-s + 1.40·73-s + 0.450·79-s − 1.67·91-s − 2.03·97-s + 1.18·103-s + 1.14·109-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(331776\)    =    \(2^{12} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(21.1543\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 331776,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.917961877\)
\(L(\frac12)\) \(\approx\) \(1.917961877\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.7.ae_o
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.e_o
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.29.a_bi
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.31.ae_ck
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.am_dq
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.a_ack
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ai_bm
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.53.a_ada
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.59.a_bm
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.61.am_fm
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.71.a_aco
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.ae_ew
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.83.a_acg
89$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.89.a_s
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−8.517435712552720540153705474063, −8.351351724267409661671391831605, −7.85114458006293673378279750292, −7.56320705414644089316355988202, −6.87777462327658070769750452915, −6.48962675486503448187587911702, −5.83363034091370664657469280060, −5.42535620906012333059112612771, −4.67694632292605577593892500874, −4.47664172615705651548387574263, −4.04502773141276547795577883139, −3.05059216519893943277892409853, −2.22835922558627995371994610312, −2.02216791177006478053953062530, −0.800595118721389354819203871787, 0.800595118721389354819203871787, 2.02216791177006478053953062530, 2.22835922558627995371994610312, 3.05059216519893943277892409853, 4.04502773141276547795577883139, 4.47664172615705651548387574263, 4.67694632292605577593892500874, 5.42535620906012333059112612771, 5.83363034091370664657469280060, 6.48962675486503448187587911702, 6.87777462327658070769750452915, 7.56320705414644089316355988202, 7.85114458006293673378279750292, 8.351351724267409661671391831605, 8.517435712552720540153705474063

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