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Properties

Label 4-739328-1.1-c1e2-0-8
Degree $4$
Conductor $739328$
Sign $-1$
Analytic cond. $47.1401$
Root an. cond. $2.62028$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·9-s + 12·13-s + 4·17-s − 6·25-s − 20·29-s − 4·37-s − 14·49-s + 28·53-s − 12·73-s + 27·81-s + 12·109-s − 72·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2·9-s + 3.32·13-s + 0.970·17-s − 6/5·25-s − 3.71·29-s − 0.657·37-s − 2·49-s + 3.84·53-s − 1.40·73-s + 3·81-s + 1.14·109-s − 6.65·117-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 − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 739328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(739328\)    =    \(2^{11} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(47.1401\)
Root analytic conductor: \(2.62028\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 739328,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
19$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.29.u_gc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
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$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.53.abc_lq
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−8.082347486973466875367987545617, −7.77199473906097062385997282225, −7.28916126993973683380301647461, −6.56558924270052789983772431797, −5.96126926568006819265365532145, −5.87146418848833687506982135026, −5.57799879021137608863398098635, −5.09050475729599485889830101555, −3.85524114672212228883256453900, −3.67478222653086463350186782835, −3.56978615589504816697349130571, −2.67167944220030971104507326816, −1.87948404419680995103462610417, −1.22433077509918601428054581575, 0, 1.22433077509918601428054581575, 1.87948404419680995103462610417, 2.67167944220030971104507326816, 3.56978615589504816697349130571, 3.67478222653086463350186782835, 3.85524114672212228883256453900, 5.09050475729599485889830101555, 5.57799879021137608863398098635, 5.87146418848833687506982135026, 5.96126926568006819265365532145, 6.56558924270052789983772431797, 7.28916126993973683380301647461, 7.77199473906097062385997282225, 8.082347486973466875367987545617

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