Properties

Label 4-1800e2-1.1-c1e2-0-31
Degree $4$
Conductor $3240000$
Sign $1$
Analytic cond. $206.585$
Root an. cond. $3.79118$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·11-s − 8·19-s − 4·29-s − 16·31-s + 12·41-s − 2·49-s − 8·59-s − 4·61-s − 12·89-s − 12·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2.41·11-s − 1.83·19-s − 0.742·29-s − 2.87·31-s + 1.87·41-s − 2/7·49-s − 1.04·59-s − 0.512·61-s − 1.27·89-s − 1.19·101-s − 2.68·109-s + 2.36·121-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 + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3240000\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(206.585\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3240000,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−8.873584566447424430706400357654, −8.847709111336608870019396406632, −8.243293411874429110793515009153, −7.77080944611726142621936812250, −7.52119316008319306292488441513, −7.41172780847449720363020333335, −6.55968730212721699227884327880, −6.36738877198213162555516852131, −5.62377263191614303710863542778, −5.52542560750406947022530639259, −5.09528650114999427237624895685, −4.58885776614249694861552563657, −3.87672902355966487323261371011, −3.85879349896762081795401640517, −2.79112454019451606856737674244, −2.67418249861115509209892080655, −2.06277990496082175008236103009, −1.48499495784257744217639413273, 0, 0, 1.48499495784257744217639413273, 2.06277990496082175008236103009, 2.67418249861115509209892080655, 2.79112454019451606856737674244, 3.85879349896762081795401640517, 3.87672902355966487323261371011, 4.58885776614249694861552563657, 5.09528650114999427237624895685, 5.52542560750406947022530639259, 5.62377263191614303710863542778, 6.36738877198213162555516852131, 6.55968730212721699227884327880, 7.41172780847449720363020333335, 7.52119316008319306292488441513, 7.77080944611726142621936812250, 8.243293411874429110793515009153, 8.847709111336608870019396406632, 8.873584566447424430706400357654

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