Properties

Label 4-10e4-1.1-c3e2-0-5
Degree $4$
Conductor $10000$
Sign $1$
Analytic cond. $34.8122$
Root an. cond. $2.42903$
Motivic weight $3$
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  + 53·9-s + 90·11-s + 182·19-s − 288·29-s + 52·31-s − 918·41-s + 10·49-s + 144·59-s − 236·61-s + 216·71-s + 1.79e3·79-s + 2.08e3·81-s − 702·89-s + 4.77e3·99-s − 1.90e3·101-s + 1.60e3·109-s + 3.41e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.45e3·169-s + ⋯
L(s)  = 1  + 1.96·9-s + 2.46·11-s + 2.19·19-s − 1.84·29-s + 0.301·31-s − 3.49·41-s + 0.0291·49-s + 0.317·59-s − 0.495·61-s + 0.361·71-s + 2.55·79-s + 2.85·81-s − 0.836·89-s + 4.84·99-s − 1.87·101-s + 1.40·109-s + 2.56·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.11·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10000\)    =    \(2^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(34.8122\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{100} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10000,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.906233202\)
\(L(\frac12)\) \(\approx\) \(2.906233202\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 53 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 - 45 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 2458 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 3863 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 91 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 24010 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 144 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 26 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 396 T + p^{3} T^{2} )( 1 + 396 T + p^{3} T^{2} ) \)
41$C_2$ \( ( 1 + 459 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 52586 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 + 11378 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 13610 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 72 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 118 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 538525 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 108 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 688633 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 898 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 284245 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 351 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 1676350 T^{2} + p^{6} T^{4} \)
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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

−13.72918646526290778297024060138, −13.23685780285049394962348753335, −12.42442482096761679933996178079, −12.12961248404188290857981544684, −11.57849437813299749432542718192, −11.21835019734493481088449009373, −10.13399282547714886175612445311, −9.949827041776075875339728072208, −9.233911830151405501417135977977, −9.099306185678696802390083690620, −7.985330032480269273994850694375, −7.38174822065435501609993085107, −6.79863395467540329750603402373, −6.53777620311412573802084483033, −5.41007124147762024486604492193, −4.74320459809182051351148795248, −3.70804106047148059952631965846, −3.63260068562254612262995119738, −1.69036601213490694776935665105, −1.20542120539560899129498037758, 1.20542120539560899129498037758, 1.69036601213490694776935665105, 3.63260068562254612262995119738, 3.70804106047148059952631965846, 4.74320459809182051351148795248, 5.41007124147762024486604492193, 6.53777620311412573802084483033, 6.79863395467540329750603402373, 7.38174822065435501609993085107, 7.985330032480269273994850694375, 9.099306185678696802390083690620, 9.233911830151405501417135977977, 9.949827041776075875339728072208, 10.13399282547714886175612445311, 11.21835019734493481088449009373, 11.57849437813299749432542718192, 12.12961248404188290857981544684, 12.42442482096761679933996178079, 13.23685780285049394962348753335, 13.72918646526290778297024060138

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