Properties

Label 4-1960e2-1.1-c1e2-0-2
Degree $4$
Conductor $3841600$
Sign $1$
Analytic cond. $244.943$
Root an. cond. $3.95609$
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  − 2·3-s − 5-s + 3·9-s − 4·11-s + 4·13-s + 2·15-s − 2·19-s + 4·23-s − 10·27-s + 20·29-s − 4·31-s + 8·33-s + 2·37-s − 8·39-s − 24·41-s − 8·43-s − 3·45-s − 4·47-s − 2·53-s + 4·55-s + 4·57-s − 10·59-s − 6·61-s − 4·65-s − 4·67-s − 8·69-s − 24·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 9-s − 1.20·11-s + 1.10·13-s + 0.516·15-s − 0.458·19-s + 0.834·23-s − 1.92·27-s + 3.71·29-s − 0.718·31-s + 1.39·33-s + 0.328·37-s − 1.28·39-s − 3.74·41-s − 1.21·43-s − 0.447·45-s − 0.583·47-s − 0.274·53-s + 0.539·55-s + 0.529·57-s − 1.30·59-s − 0.768·61-s − 0.496·65-s − 0.488·67-s − 0.963·69-s − 2.84·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3841600\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(244.943\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3841600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3459646490\)
\(L(\frac12)\) \(\approx\) \(0.3459646490\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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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

−9.737164637164440922460775479793, −8.894560463961169330410621558776, −8.394103003709783179871406074336, −8.133836720361265078082033428551, −8.094931948394374216804053641383, −7.21131493635429122891472952457, −7.02395400974049047387764823666, −6.43242723721581732518618445923, −6.36363319755879462400237017109, −5.85114260334268851153971226231, −5.19163953181779158949396316990, −4.88099877546702950533352616469, −4.84032235104455141887920943262, −4.04699960680343292029668420016, −3.62644502470734658448130307549, −2.98945410855689800002583134443, −2.71905306366609883607589813878, −1.55012403627239419765466950027, −1.40942627623808003791232793947, −0.23924823840989363834036037935, 0.23924823840989363834036037935, 1.40942627623808003791232793947, 1.55012403627239419765466950027, 2.71905306366609883607589813878, 2.98945410855689800002583134443, 3.62644502470734658448130307549, 4.04699960680343292029668420016, 4.84032235104455141887920943262, 4.88099877546702950533352616469, 5.19163953181779158949396316990, 5.85114260334268851153971226231, 6.36363319755879462400237017109, 6.43242723721581732518618445923, 7.02395400974049047387764823666, 7.21131493635429122891472952457, 8.094931948394374216804053641383, 8.133836720361265078082033428551, 8.394103003709783179871406074336, 8.894560463961169330410621558776, 9.737164637164440922460775479793

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