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Properties

Label 4-24e4-1.1-c1e2-0-3
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  − 4·5-s + 4·13-s + 4·17-s + 6·25-s − 4·29-s + 4·37-s + 12·41-s + 6·49-s − 4·53-s − 12·61-s − 16·65-s − 12·73-s − 16·85-s − 12·89-s + 12·97-s + 12·101-s − 12·109-s + 12·113-s + 6·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.10·13-s + 0.970·17-s + 6/5·25-s − 0.742·29-s + 0.657·37-s + 1.87·41-s + 6/7·49-s − 0.549·53-s − 1.53·61-s − 1.98·65-s − 1.40·73-s − 1.73·85-s − 1.27·89-s + 1.21·97-s + 1.19·101-s − 1.14·109-s + 1.12·113-s + 6/11·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-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

\(L(1)\) \(\approx\) \(1.156254268\)
\(L(\frac12)\) \(\approx\) \(1.156254268\)
\(L(\frac{3}{2})\) 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$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.e_k
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ae_w
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.19.a_ba
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.e_ba
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.a_abm
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.am_dy
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.47.a_o
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.e_ec
59$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.59.a_ag
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.m_fm
67$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \) 2.67.a_ec
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.a_aek
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.m_eo
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.83.a_bq
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.am_gk
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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

−8.498496081569373807285549191202, −8.391925588786705405617316621484, −7.70067673686940046033546918439, −7.47390666566264531130184535895, −7.23720596968336809806830163371, −6.26887733392054148669298775211, −6.02530151903108223925256828434, −5.47124076590828075518360744546, −4.69793168379277165720707433216, −4.18713263931496237526004419321, −3.85070648314156028643980276555, −3.29665856805500580653590358871, −2.74050217548656748847252330619, −1.58561781277996662118904169882, −0.64966467207572311441047422696, 0.64966467207572311441047422696, 1.58561781277996662118904169882, 2.74050217548656748847252330619, 3.29665856805500580653590358871, 3.85070648314156028643980276555, 4.18713263931496237526004419321, 4.69793168379277165720707433216, 5.47124076590828075518360744546, 6.02530151903108223925256828434, 6.26887733392054148669298775211, 7.23720596968336809806830163371, 7.47390666566264531130184535895, 7.70067673686940046033546918439, 8.391925588786705405617316621484, 8.498496081569373807285549191202

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