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Properties

Label 4-2268e2-1.1-c1e2-0-20
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  + 7-s − 3·13-s + 6·19-s + 5·25-s − 37-s − 13·43-s − 6·49-s + 22·67-s − 24·73-s − 26·79-s − 3·91-s + 9·97-s + 33·103-s − 19·109-s − 11·121-s + 127-s + 131-s + 6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7·169-s + 173-s + ⋯
L(s)  = 1  + 0.377·7-s − 0.832·13-s + 1.37·19-s + 25-s − 0.164·37-s − 1.98·43-s − 6/7·49-s + 2.68·67-s − 2.80·73-s − 2.92·79-s − 0.314·91-s + 0.913·97-s + 3.25·103-s − 1.81·109-s − 121-s + 0.0887·127-s + 0.0873·131-s + 0.520·133-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 − 0.538·169-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

\(L(1)\) \(\approx\) \(2.018563221\)
\(L(\frac12)\) \(\approx\) \(2.018563221\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.11.a_l
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.d_q
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ag_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$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.a_ach
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.b_abk
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.n_ew
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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.a_er
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.67.aw_jv
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.73.y_kf
79$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.79.ba_mp
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$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.aj_eu
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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

−9.178897769789374840228326136303, −8.694748899908788055745499889006, −8.578712919384978819319673265304, −8.019249613283833509541588126839, −7.68952909875811283598634950756, −7.25182491726101085399992354996, −6.94927096091884629168902746218, −6.64528674844189043892446630343, −6.04630490301077515619014444170, −5.58342178553037669370649361847, −5.24628324518402724872483446401, −4.78494597310027743368799352860, −4.59696857323131445477521214691, −3.93027957803695273354036887392, −3.32459527405826226300462766013, −3.04216336481633787324025528550, −2.53066720823162018393243113528, −1.78380373860135650601105111534, −1.36909543619551245611186742485, −0.50382477383887372135006834477, 0.50382477383887372135006834477, 1.36909543619551245611186742485, 1.78380373860135650601105111534, 2.53066720823162018393243113528, 3.04216336481633787324025528550, 3.32459527405826226300462766013, 3.93027957803695273354036887392, 4.59696857323131445477521214691, 4.78494597310027743368799352860, 5.24628324518402724872483446401, 5.58342178553037669370649361847, 6.04630490301077515619014444170, 6.64528674844189043892446630343, 6.94927096091884629168902746218, 7.25182491726101085399992354996, 7.68952909875811283598634950756, 8.019249613283833509541588126839, 8.578712919384978819319673265304, 8.694748899908788055745499889006, 9.178897769789374840228326136303

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