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Properties

Label 4-20e4-1.1-c1e2-0-5
Degree $4$
Conductor $160000$
Sign $1$
Analytic cond. $10.2017$
Root an. cond. $1.78718$
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  + 10·13-s + 10·17-s − 10·37-s + 16·41-s − 10·53-s − 24·61-s − 10·73-s − 9·81-s − 10·97-s + 4·101-s − 30·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 50·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2.77·13-s + 2.42·17-s − 1.64·37-s + 2.49·41-s − 1.37·53-s − 3.07·61-s − 1.17·73-s − 81-s − 1.01·97-s + 0.398·101-s − 2.82·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-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 + 3.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(160000\)    =    \(2^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(10.2017\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.056160146\)
\(L(\frac12)\) \(\approx\) \(2.056160146\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
7$C_2^2$ \( 1 + p^{2} T^{4} \) 2.7.a_a
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.13.ak_by
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ak_by
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 + p^{2} T^{4} \) 2.23.a_a
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.k_by
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.41.aq_fq
43$C_2^2$ \( 1 + p^{2} T^{4} \) 2.43.a_a
47$C_2^2$ \( 1 + p^{2} T^{4} \) 2.47.a_a
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.k_by
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.61.y_kg
67$C_2^2$ \( 1 + p^{2} T^{4} \) 2.67.a_a
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.k_by
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 + p^{2} T^{4} \) 2.83.a_a
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.k_by
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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

−11.34389581752958628359760485054, −11.04472555806920005307368319708, −10.54545353312228005749820967794, −10.38665363389198613782678020822, −9.493423312652212097735151495875, −9.368931877555665881972871881716, −8.629770810801943550226109079514, −8.394194096742075503759627841014, −7.64187653849720920038089449727, −7.61950321962881613564710904487, −6.68928335851737744391590006756, −6.17528400162867302190580578900, −5.71084731742201172430389379938, −5.52846652566638912601909530456, −4.50293949880647670934962873573, −4.00464382900555206770790773095, −3.19380337442781207459228011090, −3.15925999426796568586280927185, −1.61431408663841428470958862339, −1.13266468726042336887303247194, 1.13266468726042336887303247194, 1.61431408663841428470958862339, 3.15925999426796568586280927185, 3.19380337442781207459228011090, 4.00464382900555206770790773095, 4.50293949880647670934962873573, 5.52846652566638912601909530456, 5.71084731742201172430389379938, 6.17528400162867302190580578900, 6.68928335851737744391590006756, 7.61950321962881613564710904487, 7.64187653849720920038089449727, 8.394194096742075503759627841014, 8.629770810801943550226109079514, 9.368931877555665881972871881716, 9.493423312652212097735151495875, 10.38665363389198613782678020822, 10.54545353312228005749820967794, 11.04472555806920005307368319708, 11.34389581752958628359760485054

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