Properties

Label 4-280e2-1.1-c1e2-0-2
Degree $4$
Conductor $78400$
Sign $1$
Analytic cond. $4.99885$
Root an. cond. $1.49526$
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 + 5-s − 5·7-s + 3·9-s − 2·11-s − 15-s − 4·17-s + 2·19-s + 5·21-s − 23-s − 8·27-s + 18·29-s − 4·31-s + 2·33-s − 5·35-s − 4·37-s + 2·41-s + 18·43-s + 3·45-s + 18·49-s + 4·51-s + 10·53-s − 2·55-s − 2·57-s + 10·59-s − 9·61-s − 15·63-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.88·7-s + 9-s − 0.603·11-s − 0.258·15-s − 0.970·17-s + 0.458·19-s + 1.09·21-s − 0.208·23-s − 1.53·27-s + 3.34·29-s − 0.718·31-s + 0.348·33-s − 0.845·35-s − 0.657·37-s + 0.312·41-s + 2.74·43-s + 0.447·45-s + 18/7·49-s + 0.560·51-s + 1.37·53-s − 0.269·55-s − 0.264·57-s + 1.30·59-s − 1.15·61-s − 1.88·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(78400\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4.99885\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 78400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9534785697\)
\(L(\frac12)\) \(\approx\) \(0.9534785697\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 9 T + 20 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
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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

−12.24422165925907341851244897416, −11.79460291339202189941915277972, −10.92202508336561741962648213581, −10.62121449087915718973300606748, −10.27163361767837742736937662996, −9.630427545704760159689836614916, −9.567019877068292669335617643125, −8.865511007557193034276711723516, −8.361505793328924693910925457053, −7.48073461747122807341826914782, −7.11272494485200689354641110841, −6.62122539626389563929799094781, −6.15057758984233623909254845878, −5.72469742834559554874646809618, −5.04160543654805963469012000655, −4.28899712693718974078982480025, −3.78086943524880099446214105900, −2.84302711075337274392253178577, −2.30683945843488954610806465251, −0.77596960107209571803117472311, 0.77596960107209571803117472311, 2.30683945843488954610806465251, 2.84302711075337274392253178577, 3.78086943524880099446214105900, 4.28899712693718974078982480025, 5.04160543654805963469012000655, 5.72469742834559554874646809618, 6.15057758984233623909254845878, 6.62122539626389563929799094781, 7.11272494485200689354641110841, 7.48073461747122807341826914782, 8.361505793328924693910925457053, 8.865511007557193034276711723516, 9.567019877068292669335617643125, 9.630427545704760159689836614916, 10.27163361767837742736937662996, 10.62121449087915718973300606748, 10.92202508336561741962648213581, 11.79460291339202189941915277972, 12.24422165925907341851244897416

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