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Properties

Label 4-864e2-1.1-c1e2-0-21
Degree $4$
Conductor $746496$
Sign $1$
Analytic cond. $47.5972$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·7-s + 12·19-s − 9·25-s + 18·31-s − 4·37-s + 12·43-s + 13·49-s + 16·61-s + 12·67-s + 18·73-s − 18·97-s − 24·103-s − 36·109-s − 13·121-s + 127-s + 131-s + 72·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s − 54·175-s + ⋯
L(s)  = 1  + 2.26·7-s + 2.75·19-s − 9/5·25-s + 3.23·31-s − 0.657·37-s + 1.82·43-s + 13/7·49-s + 2.04·61-s + 1.46·67-s + 2.10·73-s − 1.82·97-s − 2.36·103-s − 3.44·109-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 6.24·133-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 − 2·169-s + 0.0760·173-s − 4.08·175-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(746496\)    =    \(2^{10} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(47.5972\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 746496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.271969761\)
\(L(\frac12)\) \(\approx\) \(3.271969761\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.7.ag_x
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.31.as_fn
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.43.am_es
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.53.a_acl
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.67.am_go
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.73.as_it
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.83.a_gb
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.a_as
97$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.97.s_kp
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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

−8.180464945333468393072503118033, −7.82476086711398995912338456757, −7.75120135592916568614052087932, −6.93499396476405803707004808409, −6.65325213254225917768927617170, −5.81482520780945663550334444725, −5.45678967002093979727974577207, −5.03134570361553185426866134499, −4.78635677042327797457152601343, −3.94091986861202608778869492074, −3.77979388504550471995644973896, −2.63884396406011957309091091690, −2.42238256260612007081644598630, −1.30229225574123782841215729642, −1.11568290382473683806344984693, 1.11568290382473683806344984693, 1.30229225574123782841215729642, 2.42238256260612007081644598630, 2.63884396406011957309091091690, 3.77979388504550471995644973896, 3.94091986861202608778869492074, 4.78635677042327797457152601343, 5.03134570361553185426866134499, 5.45678967002093979727974577207, 5.81482520780945663550334444725, 6.65325213254225917768927617170, 6.93499396476405803707004808409, 7.75120135592916568614052087932, 7.82476086711398995912338456757, 8.180464945333468393072503118033

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