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Properties

Label 4-2268e2-1.1-c1e2-0-53
Degree $4$
Conductor $5143824$
Sign $1$
Analytic cond. $327.974$
Root an. cond. $4.25559$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 5·13-s + 19-s − 10·25-s − 11·31-s − 11·37-s + 13·43-s − 6·49-s − 14·61-s − 5·67-s − 17·73-s − 17·79-s + 5·91-s − 14·97-s − 26·103-s + 19·109-s − 22·121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.38·13-s + 0.229·19-s − 2·25-s − 1.97·31-s − 1.80·37-s + 1.98·43-s − 6/7·49-s − 1.79·61-s − 0.610·67-s − 1.98·73-s − 1.91·79-s + 0.524·91-s − 1.42·97-s − 2.56·103-s + 1.81·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s − 0.0867·133-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5143824\)    =    \(2^{4} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(327.974\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5143824,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.f_m
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.ab_as
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.29.a_abd
31$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.l_dm
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.l_dg
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.43.an_ew
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.53.a_acb
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.o_ff
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.f_abq
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.r_ii
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.r_ic
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.o_dv
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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

−8.735337316128786531026605872727, −8.671131994267711853551069541782, −7.916986045840507899519835116874, −7.47543495519043085394123869258, −7.36288626373026552746016215448, −7.18168453819984626874993426681, −6.28794468224460445429142665671, −6.23606275897716581455959955764, −5.50973610260415290305082869205, −5.42300451138715627894427055189, −4.90347528199009248676168407541, −4.25634642556941046501974163839, −3.99133648412656013929948771383, −3.52926385911289743124619338202, −2.81474833742590553500052434650, −2.62253147020274484765712305503, −1.73243019184440802996069156400, −1.54411190143584267816944787932, 0, 0, 1.54411190143584267816944787932, 1.73243019184440802996069156400, 2.62253147020274484765712305503, 2.81474833742590553500052434650, 3.52926385911289743124619338202, 3.99133648412656013929948771383, 4.25634642556941046501974163839, 4.90347528199009248676168407541, 5.42300451138715627894427055189, 5.50973610260415290305082869205, 6.23606275897716581455959955764, 6.28794468224460445429142665671, 7.18168453819984626874993426681, 7.36288626373026552746016215448, 7.47543495519043085394123869258, 7.916986045840507899519835116874, 8.671131994267711853551069541782, 8.735337316128786531026605872727

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