Properties

Label 4-7360e2-1.1-c1e2-0-17
Degree $4$
Conductor $54169600$
Sign $1$
Analytic cond. $3453.90$
Root an. cond. $7.66615$
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  + 3-s + 2·5-s + 7-s − 3·11-s − 7·13-s + 2·15-s − 3·17-s − 7·19-s + 21-s + 2·23-s + 3·25-s + 2·27-s − 6·29-s + 7·31-s − 3·33-s + 2·35-s + 8·37-s − 7·39-s − 9·41-s − 4·43-s − 18·47-s − 8·49-s − 3·51-s − 12·53-s − 6·55-s − 7·57-s + 18·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.377·7-s − 0.904·11-s − 1.94·13-s + 0.516·15-s − 0.727·17-s − 1.60·19-s + 0.218·21-s + 0.417·23-s + 3/5·25-s + 0.384·27-s − 1.11·29-s + 1.25·31-s − 0.522·33-s + 0.338·35-s + 1.31·37-s − 1.12·39-s − 1.40·41-s − 0.609·43-s − 2.62·47-s − 8/7·49-s − 0.420·51-s − 1.64·53-s − 0.809·55-s − 0.927·57-s + 2.34·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(54169600\)    =    \(2^{12} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(3453.90\)
Root analytic conductor: \(7.66615\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7360} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 54169600,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 7 T + 27 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 9 T + 97 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 18 T + 154 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 18 T + 178 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 7 T + 87 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 97 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 7 T + 75 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
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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

−7.72559270071860145983813959721, −7.61526191631514262568548275866, −6.90538024139718686630943959359, −6.59509008933691159367212668262, −6.45618258307858548893260869478, −6.21936048580436766920642169472, −5.33093501242736514766254819710, −5.27311148137433832417669718019, −4.92224953166670623042522099035, −4.59617821246778085282063405507, −4.34987330605809724263098116840, −3.66439985493917045119517114312, −3.13068602656314140045396435743, −2.89714219406380657439587058079, −2.26238286471411208383711708274, −2.24434515295162159333032384585, −1.79253750151979273035453353504, −1.15522517300381585841096040172, 0, 0, 1.15522517300381585841096040172, 1.79253750151979273035453353504, 2.24434515295162159333032384585, 2.26238286471411208383711708274, 2.89714219406380657439587058079, 3.13068602656314140045396435743, 3.66439985493917045119517114312, 4.34987330605809724263098116840, 4.59617821246778085282063405507, 4.92224953166670623042522099035, 5.27311148137433832417669718019, 5.33093501242736514766254819710, 6.21936048580436766920642169472, 6.45618258307858548893260869478, 6.59509008933691159367212668262, 6.90538024139718686630943959359, 7.61526191631514262568548275866, 7.72559270071860145983813959721

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