Properties

Label 4-1400e2-1.1-c1e2-0-17
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 + 3·9-s + 7·11-s − 3·13-s − 5·17-s + 2·19-s + 2·21-s − 2·23-s + 8·27-s − 3·29-s − 16·31-s + 7·33-s + 4·37-s − 3·39-s + 2·41-s + 6·43-s − 3·47-s + 3·49-s − 5·51-s − 10·53-s + 2·57-s + 16·59-s + 6·61-s + 6·63-s + 8·67-s − 2·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 9-s + 2.11·11-s − 0.832·13-s − 1.21·17-s + 0.458·19-s + 0.436·21-s − 0.417·23-s + 1.53·27-s − 0.557·29-s − 2.87·31-s + 1.21·33-s + 0.657·37-s − 0.480·39-s + 0.312·41-s + 0.914·43-s − 0.437·47-s + 3/7·49-s − 0.700·51-s − 1.37·53-s + 0.264·57-s + 2.08·59-s + 0.768·61-s + 0.755·63-s + 0.977·67-s − 0.240·69-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

\(L(1)\) \(\approx\) \(3.724672027\)
\(L(\frac12)\) \(\approx\) \(3.724672027\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 13 T + 126 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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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

−9.522499361704848014184694646345, −9.397991707027163850618833447413, −9.109179329456040175650741661490, −8.599660450537003361665691671205, −8.189525083927286659334455294686, −7.72950235521558262835445175074, −7.34381066193211787037707130626, −6.91045554305168647134119697223, −6.63020756891676730928343959662, −6.28683622552320494088663864291, −5.43099998452876702819854300790, −5.18357888397915296882654963675, −4.61292101077022212155390767498, −4.09603780144951542613436311843, −3.79847429204527997960070107939, −3.48083297514032757813094558731, −2.38984346767194484336583609876, −2.11361203192552443151639818170, −1.54427704093114398314754695480, −0.800014161814407391550046830876, 0.800014161814407391550046830876, 1.54427704093114398314754695480, 2.11361203192552443151639818170, 2.38984346767194484336583609876, 3.48083297514032757813094558731, 3.79847429204527997960070107939, 4.09603780144951542613436311843, 4.61292101077022212155390767498, 5.18357888397915296882654963675, 5.43099998452876702819854300790, 6.28683622552320494088663864291, 6.63020756891676730928343959662, 6.91045554305168647134119697223, 7.34381066193211787037707130626, 7.72950235521558262835445175074, 8.189525083927286659334455294686, 8.599660450537003361665691671205, 9.109179329456040175650741661490, 9.397991707027163850618833447413, 9.522499361704848014184694646345

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