Properties

Label 4-5070e2-1.1-c1e2-0-0
Degree $4$
Conductor $25704900$
Sign $1$
Analytic cond. $1638.96$
Root an. cond. $6.36271$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 3·4-s − 2·5-s + 4·6-s − 4·7-s − 4·8-s + 3·9-s + 4·10-s − 6·11-s − 6·12-s + 8·14-s + 4·15-s + 5·16-s + 8·17-s − 6·18-s − 8·19-s − 6·20-s + 8·21-s + 12·22-s − 4·23-s + 8·24-s + 3·25-s − 4·27-s − 12·28-s + 4·29-s − 8·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 1.51·7-s − 1.41·8-s + 9-s + 1.26·10-s − 1.80·11-s − 1.73·12-s + 2.13·14-s + 1.03·15-s + 5/4·16-s + 1.94·17-s − 1.41·18-s − 1.83·19-s − 1.34·20-s + 1.74·21-s + 2.55·22-s − 0.834·23-s + 1.63·24-s + 3/5·25-s − 0.769·27-s − 2.26·28-s + 0.742·29-s − 1.46·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25704900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1638.96\)
Root analytic conductor: \(6.36271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25704900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1095075535\)
\(L(\frac12)\) \(\approx\) \(0.1095075535\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 47 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 4 T + 59 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 63 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 14 T + 131 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 10 T + 131 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 14 T + 159 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 122 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 8 T + 182 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 198 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−8.206222191889473209321876517812, −8.121286495555366771519048716328, −7.62228499162713729948991963866, −7.61366979273131513139950851041, −6.84874120735922990869744821052, −6.74490023567640168181283829848, −6.29262852329280302628441366762, −6.17275304526089447714995268451, −5.46148624164077495930780017736, −5.43312739106038741166964419959, −4.67157049843353134107021888722, −4.57938699461460910879882299454, −3.58256311170217966191857119843, −3.57401534631699591975891260305, −2.89745681348894234828070251987, −2.78920108007203081999506844165, −1.78767712833647801329879424957, −1.60911771537227010463981368190, −0.53771870815811906812815603358, −0.23165057481311176227041662596, 0.23165057481311176227041662596, 0.53771870815811906812815603358, 1.60911771537227010463981368190, 1.78767712833647801329879424957, 2.78920108007203081999506844165, 2.89745681348894234828070251987, 3.57401534631699591975891260305, 3.58256311170217966191857119843, 4.57938699461460910879882299454, 4.67157049843353134107021888722, 5.43312739106038741166964419959, 5.46148624164077495930780017736, 6.17275304526089447714995268451, 6.29262852329280302628441366762, 6.74490023567640168181283829848, 6.84874120735922990869744821052, 7.61366979273131513139950851041, 7.62228499162713729948991963866, 8.121286495555366771519048716328, 8.206222191889473209321876517812

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