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Properties

Label 4-2268e2-1.1-c1e2-0-33
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·7-s + 9·13-s − 6·19-s + 5·25-s + 11·37-s + 5·43-s + 18·49-s + 10·67-s + 24·73-s + 34·79-s − 45·91-s + 33·97-s + 27·103-s + 17·109-s − 11·121-s + 127-s + 131-s + 30·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 41·169-s + 173-s + ⋯
L(s)  = 1  − 1.88·7-s + 2.49·13-s − 1.37·19-s + 25-s + 1.80·37-s + 0.762·43-s + 18/7·49-s + 1.22·67-s + 2.80·73-s + 3.82·79-s − 4.71·91-s + 3.35·97-s + 2.66·103-s + 1.62·109-s − 121-s + 0.0887·127-s + 0.0873·131-s + 2.60·133-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 + 3.15·169-s + 0.0760·173-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: \(0\)
Selberg data: \((4,\ 5143824,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.462179974\)
\(L(\frac12)\) \(\approx\) \(2.462179974\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.11.a_l
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.aj_bo
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.g_bf
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.23.a_x
29$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.29.a_bd
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.a_n
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.37.al_dg
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.af_as
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.53.a_cb
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.a_abv
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.67.ak_gd
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.73.ay_kf
79$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \) 2.79.abi_rf
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$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 - 14 T + p T^{2} ) \) 2.97.abh_rs
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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

−9.044358096193202531950376457754, −9.020817071667325867331538063800, −8.348275267540610850365690019052, −8.342510566650235392274931561131, −7.59369135816575395933692367868, −7.34725534663125750100354162655, −6.49941174506660859787648970768, −6.46415324043676029073908775479, −6.19372376651050154847371255375, −6.07425962925085096302905983249, −5.22730256120617510943540783306, −4.92016125897948225393607489992, −4.02339696029836918657524162888, −4.00951257981786266515122698639, −3.33135255046685282101392931891, −3.29551840786823861688108029193, −2.32958565048850864888792223325, −2.15277405916169991721565000997, −0.880156496642688404910303371621, −0.75561524433014092291946825263, 0.75561524433014092291946825263, 0.880156496642688404910303371621, 2.15277405916169991721565000997, 2.32958565048850864888792223325, 3.29551840786823861688108029193, 3.33135255046685282101392931891, 4.00951257981786266515122698639, 4.02339696029836918657524162888, 4.92016125897948225393607489992, 5.22730256120617510943540783306, 6.07425962925085096302905983249, 6.19372376651050154847371255375, 6.46415324043676029073908775479, 6.49941174506660859787648970768, 7.34725534663125750100354162655, 7.59369135816575395933692367868, 8.342510566650235392274931561131, 8.348275267540610850365690019052, 9.020817071667325867331538063800, 9.044358096193202531950376457754

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