Properties

Label 4-5290e2-1.1-c1e2-0-0
Degree $4$
Conductor $27984100$
Sign $1$
Analytic cond. $1784.29$
Root an. cond. $6.49929$
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 + 3·3-s + 3·4-s − 2·5-s − 6·6-s − 3·7-s − 4·8-s + 4·9-s + 4·10-s + 7·11-s + 9·12-s + 3·13-s + 6·14-s − 6·15-s + 5·16-s − 3·17-s − 8·18-s − 19-s − 6·20-s − 9·21-s − 14·22-s − 12·24-s + 3·25-s − 6·26-s + 6·27-s − 9·28-s + 2·29-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.73·3-s + 3/2·4-s − 0.894·5-s − 2.44·6-s − 1.13·7-s − 1.41·8-s + 4/3·9-s + 1.26·10-s + 2.11·11-s + 2.59·12-s + 0.832·13-s + 1.60·14-s − 1.54·15-s + 5/4·16-s − 0.727·17-s − 1.88·18-s − 0.229·19-s − 1.34·20-s − 1.96·21-s − 2.98·22-s − 2.44·24-s + 3/5·25-s − 1.17·26-s + 1.15·27-s − 1.70·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27984100\)    =    \(2^{2} \cdot 5^{2} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(1784.29\)
Root analytic conductor: \(6.49929\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 27984100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9874829435\)
\(L(\frac12)\) \(\approx\) \(0.9874829435\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
23 \( 1 \)
good3$C_4$ \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 7 T + 31 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 5 T + 47 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 29 T + 349 T^{2} + 29 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 9 T + 211 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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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

−8.477419287164700114095098197678, −8.248573402024787708120180275294, −7.76724255537707758042311995469, −7.42858141320631188034334227164, −6.98661838218681157121995456683, −6.70518536026287629508048521555, −6.56884801584253344478848150125, −6.25429186750843495409820929880, −5.60849499054555336226489193568, −5.13026902404799126415466140273, −4.29247548357718502110116069873, −4.14976797084711345520746040983, −3.64697220090040223162911819007, −3.39399312264718667133158518062, −2.99727562464875189347740904144, −2.78761518806310236798459870958, −1.83096084331165538535398668178, −1.73148530922489940735699376701, −1.17774502232057396099768266948, −0.30402522081653973755426818255, 0.30402522081653973755426818255, 1.17774502232057396099768266948, 1.73148530922489940735699376701, 1.83096084331165538535398668178, 2.78761518806310236798459870958, 2.99727562464875189347740904144, 3.39399312264718667133158518062, 3.64697220090040223162911819007, 4.14976797084711345520746040983, 4.29247548357718502110116069873, 5.13026902404799126415466140273, 5.60849499054555336226489193568, 6.25429186750843495409820929880, 6.56884801584253344478848150125, 6.70518536026287629508048521555, 6.98661838218681157121995456683, 7.42858141320631188034334227164, 7.76724255537707758042311995469, 8.248573402024787708120180275294, 8.477419287164700114095098197678

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