Properties

Label 4-2475e2-1.1-c1e2-0-22
Degree $4$
Conductor $6125625$
Sign $1$
Analytic cond. $390.575$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4-s − 2·7-s + 2·11-s + 4·14-s + 16-s − 2·17-s + 10·19-s − 4·22-s − 2·23-s − 2·28-s − 8·29-s + 2·32-s + 4·34-s − 6·37-s − 20·38-s + 2·41-s − 12·43-s + 2·44-s + 4·46-s − 2·47-s − 9·49-s − 4·53-s + 16·58-s − 22·59-s + 12·61-s − 11·64-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s − 0.755·7-s + 0.603·11-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 2.29·19-s − 0.852·22-s − 0.417·23-s − 0.377·28-s − 1.48·29-s + 0.353·32-s + 0.685·34-s − 0.986·37-s − 3.24·38-s + 0.312·41-s − 1.82·43-s + 0.301·44-s + 0.589·46-s − 0.291·47-s − 9/7·49-s − 0.549·53-s + 2.10·58-s − 2.86·59-s + 1.53·61-s − 1.37·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6125625\)    =    \(3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(390.575\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6125625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 10 T + 61 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 75 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 23 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 12 T + 138 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 10 T + 159 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 18 T + 221 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 195 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−8.678851104420633197157236405708, −8.625642048001207447011655012688, −7.921733426296923945198293828549, −7.76979342236045715315367749092, −7.37438829940925561678521700520, −6.85688251786561387750474875400, −6.60876097217849553685173723197, −6.13685268945600715414667838622, −5.62922886800454256888931881899, −5.40669880609907489276588961518, −4.61635195213835129702992901921, −4.51768495835261361219229922652, −3.57779519582076510931555663776, −3.32100574000471177215824328060, −3.09742118113987911399791879819, −2.25141443486313841659232461764, −1.41352342089987355386615392530, −1.32380974880041395191507705640, 0, 0, 1.32380974880041395191507705640, 1.41352342089987355386615392530, 2.25141443486313841659232461764, 3.09742118113987911399791879819, 3.32100574000471177215824328060, 3.57779519582076510931555663776, 4.51768495835261361219229922652, 4.61635195213835129702992901921, 5.40669880609907489276588961518, 5.62922886800454256888931881899, 6.13685268945600715414667838622, 6.60876097217849553685173723197, 6.85688251786561387750474875400, 7.37438829940925561678521700520, 7.76979342236045715315367749092, 7.921733426296923945198293828549, 8.625642048001207447011655012688, 8.678851104420633197157236405708

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