Properties

Label 4-1400e2-1.1-c1e2-0-14
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·7-s − 9-s − 4·11-s + 4·13-s − 17-s + 3·19-s + 2·21-s + 7·23-s + 11·29-s − 10·31-s − 4·33-s − 3·37-s + 4·39-s − 9·41-s + 9·43-s + 3·49-s − 51-s + 6·53-s + 3·57-s − 2·59-s + 14·61-s − 2·63-s + 24·67-s + 7·69-s − 9·71-s + 17·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s − 1/3·9-s − 1.20·11-s + 1.10·13-s − 0.242·17-s + 0.688·19-s + 0.436·21-s + 1.45·23-s + 2.04·29-s − 1.79·31-s − 0.696·33-s − 0.493·37-s + 0.640·39-s − 1.40·41-s + 1.37·43-s + 3/7·49-s − 0.140·51-s + 0.824·53-s + 0.397·57-s − 0.260·59-s + 1.79·61-s − 0.251·63-s + 2.93·67-s + 0.842·69-s − 1.06·71-s + 1.98·73-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: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.120201953\)
\(L(\frac12)\) \(\approx\) \(3.120201953\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 3 T + 36 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 11 T + 84 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 3 T - 30 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 9 T + 98 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 9 T + 102 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T - 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 24 T + 261 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 9 T + 124 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 17 T + 214 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - T + 120 T^{2} - p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 15 T + 184 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + T + 72 T^{2} + p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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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

−9.764886541843858359071372728501, −9.218237792899048326957211802772, −8.744118174692972566389473194700, −8.623794184886935393997146175628, −8.239847596105057719892657208289, −7.928524300526593613706540030669, −7.20296901163723082148638127686, −7.16615429425786812669629194646, −6.57094489271139272959239945950, −6.03755799412375172387116788170, −5.38190326402091152015642817625, −5.24799940672053052538104844065, −4.85377048861373154030166071134, −4.21077700994101779347989228616, −3.44642196103307571227694096033, −3.41357399021207989540056396406, −2.46665012129860963430011425279, −2.37833347873945477786681427203, −1.37875726556887355406836628188, −0.74789096578835579845080643150, 0.74789096578835579845080643150, 1.37875726556887355406836628188, 2.37833347873945477786681427203, 2.46665012129860963430011425279, 3.41357399021207989540056396406, 3.44642196103307571227694096033, 4.21077700994101779347989228616, 4.85377048861373154030166071134, 5.24799940672053052538104844065, 5.38190326402091152015642817625, 6.03755799412375172387116788170, 6.57094489271139272959239945950, 7.16615429425786812669629194646, 7.20296901163723082148638127686, 7.928524300526593613706540030669, 8.239847596105057719892657208289, 8.623794184886935393997146175628, 8.744118174692972566389473194700, 9.218237792899048326957211802772, 9.764886541843858359071372728501

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