Properties

Label 4-1900e2-1.1-c1e2-0-6
Degree $4$
Conductor $3610000$
Sign $1$
Analytic cond. $230.176$
Root an. cond. $3.89507$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·9-s − 8·11-s − 13-s + 3·17-s − 8·19-s + 5·23-s − 8·27-s − 7·29-s + 8·31-s + 8·33-s − 20·37-s + 39-s + 5·41-s − 5·43-s − 7·47-s − 14·49-s − 3·51-s + 11·53-s + 8·57-s − 3·59-s − 11·61-s − 3·67-s − 5·69-s − 11·71-s + 15·73-s + 13·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 9-s − 2.41·11-s − 0.277·13-s + 0.727·17-s − 1.83·19-s + 1.04·23-s − 1.53·27-s − 1.29·29-s + 1.43·31-s + 1.39·33-s − 3.28·37-s + 0.160·39-s + 0.780·41-s − 0.762·43-s − 1.02·47-s − 2·49-s − 0.420·51-s + 1.51·53-s + 1.05·57-s − 0.390·59-s − 1.40·61-s − 0.366·67-s − 0.601·69-s − 1.30·71-s + 1.75·73-s + 1.46·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3610000\)    =    \(2^{4} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(230.176\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3610000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 5 T + 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 11 T + 50 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 15 T + 152 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−8.989845845342367437544805239775, −8.622435830875364603478764477064, −7.989570482257657290675379671721, −7.892524891917832966917134819022, −7.59791217356223092350716492494, −6.98322702247328319463089078027, −6.55663909393840842003513273988, −6.45199098482668909269739373495, −5.53763941380112885393628704764, −5.40615849177093872873670141514, −5.00032343592065568304053243122, −4.75450861290282967168848324743, −3.99667377812366721334553729347, −3.68531751020570680582832074051, −2.91987037007810720725395987440, −2.60799685690575793086005394328, −1.84338344060154728953582166081, −1.45625333627236327551263550352, 0, 0, 1.45625333627236327551263550352, 1.84338344060154728953582166081, 2.60799685690575793086005394328, 2.91987037007810720725395987440, 3.68531751020570680582832074051, 3.99667377812366721334553729347, 4.75450861290282967168848324743, 5.00032343592065568304053243122, 5.40615849177093872873670141514, 5.53763941380112885393628704764, 6.45199098482668909269739373495, 6.55663909393840842003513273988, 6.98322702247328319463089078027, 7.59791217356223092350716492494, 7.892524891917832966917134819022, 7.989570482257657290675379671721, 8.622435830875364603478764477064, 8.989845845342367437544805239775

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