Properties

Label 4-1400e2-1.1-c1e2-0-36
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $124.971$
Root an. cond. $3.34350$
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  − 3-s − 2·7-s − 9-s − 11-s − 13-s − 11·17-s − 6·19-s + 2·21-s + 2·23-s + 5·29-s − 4·31-s + 33-s + 39-s + 6·41-s + 6·43-s − 9·47-s + 3·49-s + 11·51-s + 18·53-s + 6·57-s − 8·59-s − 22·61-s + 2·63-s + 12·67-s − 2·69-s − 8·73-s + 2·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s − 1/3·9-s − 0.301·11-s − 0.277·13-s − 2.66·17-s − 1.37·19-s + 0.436·21-s + 0.417·23-s + 0.928·29-s − 0.718·31-s + 0.174·33-s + 0.160·39-s + 0.937·41-s + 0.914·43-s − 1.31·47-s + 3/7·49-s + 1.54·51-s + 2.47·53-s + 0.794·57-s − 1.04·59-s − 2.81·61-s + 0.251·63-s + 1.46·67-s − 0.240·69-s − 0.936·73-s + 0.227·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(124.971\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1960000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 9 T + 110 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 22 T + 226 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 11 T + 150 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 15 T + 212 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
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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

−9.167696575589424661994937861020, −9.098482402185844949142281072268, −8.486227933142193552592465662356, −8.449478658662268270449685917465, −7.64061143026776784073651096175, −7.23744510400946632512343354789, −6.74956585590966523874038437379, −6.58363209623201446301267933068, −6.03279049860851719074815280997, −5.79246375377138578107140330272, −5.18923717007332597449709250332, −4.63916836026522710379661577095, −4.18158383396793899454387122321, −4.06261651262310558125004564964, −3.03602382277812977809384611934, −2.61990222084121645000190778395, −2.25710122632834324021120631193, −1.38624561847245772024537446681, 0, 0, 1.38624561847245772024537446681, 2.25710122632834324021120631193, 2.61990222084121645000190778395, 3.03602382277812977809384611934, 4.06261651262310558125004564964, 4.18158383396793899454387122321, 4.63916836026522710379661577095, 5.18923717007332597449709250332, 5.79246375377138578107140330272, 6.03279049860851719074815280997, 6.58363209623201446301267933068, 6.74956585590966523874038437379, 7.23744510400946632512343354789, 7.64061143026776784073651096175, 8.449478658662268270449685917465, 8.486227933142193552592465662356, 9.098482402185844949142281072268, 9.167696575589424661994937861020

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