Properties

Label 4-770e2-1.1-c1e2-0-24
Degree $4$
Conductor $592900$
Sign $1$
Analytic cond. $37.8038$
Root an. cond. $2.47961$
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 + 4·3-s + 3·4-s + 2·5-s + 8·6-s + 2·7-s + 4·8-s + 6·9-s + 4·10-s − 2·11-s + 12·12-s + 2·13-s + 4·14-s + 8·15-s + 5·16-s − 2·17-s + 12·18-s − 2·19-s + 6·20-s + 8·21-s − 4·22-s − 2·23-s + 16·24-s + 3·25-s + 4·26-s − 4·27-s + 6·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 3/2·4-s + 0.894·5-s + 3.26·6-s + 0.755·7-s + 1.41·8-s + 2·9-s + 1.26·10-s − 0.603·11-s + 3.46·12-s + 0.554·13-s + 1.06·14-s + 2.06·15-s + 5/4·16-s − 0.485·17-s + 2.82·18-s − 0.458·19-s + 1.34·20-s + 1.74·21-s − 0.852·22-s − 0.417·23-s + 3.26·24-s + 3/5·25-s + 0.784·26-s − 0.769·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(592900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(37.8038\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{770} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 592900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(12.90188090\)
\(L(\frac12)\) \(\approx\) \(12.90188090\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 14 T + 122 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
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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

−10.55348405566603840281525483099, −10.14783602097039943396055759284, −9.507044163423851554345460825530, −9.259740009163592005687544437399, −8.589571810335369186252568363816, −8.422076705731661692957613300515, −8.094738857572422564011297863893, −7.57892340876491415450271085909, −7.11335892828630119795790234068, −6.50910631703067432125052856275, −6.11423148148720500758859467564, −5.52216342191513768986411681788, −5.06059656390981273772252475504, −4.61521477311126278475137722645, −3.87372146782149442686012785408, −3.58510317006896296689691358946, −2.84702398510975616674844560252, −2.67990546288820200222337348371, −1.83323675261394832793354842974, −1.77929909639200479892275302243, 1.77929909639200479892275302243, 1.83323675261394832793354842974, 2.67990546288820200222337348371, 2.84702398510975616674844560252, 3.58510317006896296689691358946, 3.87372146782149442686012785408, 4.61521477311126278475137722645, 5.06059656390981273772252475504, 5.52216342191513768986411681788, 6.11423148148720500758859467564, 6.50910631703067432125052856275, 7.11335892828630119795790234068, 7.57892340876491415450271085909, 8.094738857572422564011297863893, 8.422076705731661692957613300515, 8.589571810335369186252568363816, 9.259740009163592005687544437399, 9.507044163423851554345460825530, 10.14783602097039943396055759284, 10.55348405566603840281525483099

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