Properties

Label 4-1100e2-1.1-c1e2-0-1
Degree $4$
Conductor $1210000$
Sign $1$
Analytic cond. $77.1506$
Root an. cond. $2.96370$
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  + 5·9-s − 2·11-s − 16·19-s + 10·31-s + 10·49-s − 6·59-s − 8·61-s + 30·71-s − 4·79-s + 16·81-s + 18·89-s − 10·99-s + 36·101-s − 4·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 80·171-s + 173-s + ⋯
L(s)  = 1  + 5/3·9-s − 0.603·11-s − 3.67·19-s + 1.79·31-s + 10/7·49-s − 0.781·59-s − 1.02·61-s + 3.56·71-s − 0.450·79-s + 16/9·81-s + 1.90·89-s − 1.00·99-s + 3.58·101-s − 0.383·109-s + 3/11·121-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 + 0.769·169-s − 6.11·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1210000\)    =    \(2^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(77.1506\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1210000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.865468372\)
\(L(\frac12)\) \(\approx\) \(1.865468372\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 133 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
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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

−10.01313492178329116891562054435, −9.940319271789387594922043631822, −9.051961255465652894133445667175, −8.946503632090740994143460853420, −8.351345440658039369721313226047, −8.062373686194712659345625370078, −7.55982586091152438721473719724, −7.21803553558152379474616634641, −6.44007201054515425127993892638, −6.43930632135198331221780231950, −6.13009207437561717601615796590, −5.17527300427536703996650957480, −4.71045373961920044703539823180, −4.49530473247167458712421475269, −3.93062226249658557115565219678, −3.58388138069755241678952823937, −2.38937898927773957860052164305, −2.37997536664755934843867088895, −1.58400607519580161411540482929, −0.61015075114485358519503579589, 0.61015075114485358519503579589, 1.58400607519580161411540482929, 2.37997536664755934843867088895, 2.38937898927773957860052164305, 3.58388138069755241678952823937, 3.93062226249658557115565219678, 4.49530473247167458712421475269, 4.71045373961920044703539823180, 5.17527300427536703996650957480, 6.13009207437561717601615796590, 6.43930632135198331221780231950, 6.44007201054515425127993892638, 7.21803553558152379474616634641, 7.55982586091152438721473719724, 8.062373686194712659345625370078, 8.351345440658039369721313226047, 8.946503632090740994143460853420, 9.051961255465652894133445667175, 9.940319271789387594922043631822, 10.01313492178329116891562054435

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