Properties

Label 4-1400e2-1.1-c1e2-0-37
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 − 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: \(2\)
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$D_{4}$ \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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

−9.448236076997645326736019268313, −9.033708797379876512708220429216, −8.401799056632564739812342131172, −8.268059032052610293781751519605, −7.61480954278663796215052073954, −7.36629948391706331494942942697, −6.99799545991324445237482823730, −6.33825622347197078298359905109, −6.09437465928392248430617542934, −5.62455553357252696041882047771, −5.09436554838127089011038518997, −4.97851328967325605875195873002, −4.29528314099125389739771012556, −3.71589176684788971855901289193, −3.00158667861434916034148995806, −2.87557156144308720123594311588, −2.12815661854336525576524926019, −1.39586011083815699631381559088, 0, 0, 1.39586011083815699631381559088, 2.12815661854336525576524926019, 2.87557156144308720123594311588, 3.00158667861434916034148995806, 3.71589176684788971855901289193, 4.29528314099125389739771012556, 4.97851328967325605875195873002, 5.09436554838127089011038518997, 5.62455553357252696041882047771, 6.09437465928392248430617542934, 6.33825622347197078298359905109, 6.99799545991324445237482823730, 7.36629948391706331494942942697, 7.61480954278663796215052073954, 8.268059032052610293781751519605, 8.401799056632564739812342131172, 9.033708797379876512708220429216, 9.448236076997645326736019268313

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