Properties

Label 4-7360e2-1.1-c1e2-0-0
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 7-s − 9-s − 4·11-s + 3·13-s + 2·15-s + 17-s − 12·19-s − 21-s + 2·23-s + 3·25-s − 4·29-s − 2·31-s + 4·33-s − 2·35-s + 5·37-s − 3·39-s + 2·45-s − 3·47-s − 9·49-s − 51-s + 5·53-s + 8·55-s + 12·57-s − 5·59-s + 6·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 0.377·7-s − 1/3·9-s − 1.20·11-s + 0.832·13-s + 0.516·15-s + 0.242·17-s − 2.75·19-s − 0.218·21-s + 0.417·23-s + 3/5·25-s − 0.742·29-s − 0.359·31-s + 0.696·33-s − 0.338·35-s + 0.821·37-s − 0.480·39-s + 0.298·45-s − 0.437·47-s − 9/7·49-s − 0.140·51-s + 0.686·53-s + 1.07·55-s + 1.58·57-s − 0.650·59-s + 0.768·61-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: \(0\)
Selberg data: \((4,\ 54169600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3590737157\)
\(L(\frac12)\) \(\approx\) \(0.3590737157\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 5 T + 76 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 5 T + 86 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 6 T + 114 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 13 T + 172 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 161 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 13 T + 184 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 18 T + 258 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
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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

−8.096683657834363950212778636406, −7.80337201638930396026904987793, −7.53646783731309843545785946743, −6.90705818757530641793191366558, −6.55186184829324675613080525884, −6.55149312304631036741604650053, −5.82523467346732724361261848658, −5.77083966711572480116835402839, −5.12464647385414978552669869920, −5.03750969434301862917920422395, −4.53166996611470291734280220608, −4.13561623935012745516021747789, −3.70876848149490520475873350230, −3.61407497452552049798978077256, −2.72982507247659226570294411801, −2.63418107170488340334633108605, −1.99846768591812328884215442760, −1.59020388430224735192338928215, −0.806080642201332608709624620064, −0.19297313723472051299983867567, 0.19297313723472051299983867567, 0.806080642201332608709624620064, 1.59020388430224735192338928215, 1.99846768591812328884215442760, 2.63418107170488340334633108605, 2.72982507247659226570294411801, 3.61407497452552049798978077256, 3.70876848149490520475873350230, 4.13561623935012745516021747789, 4.53166996611470291734280220608, 5.03750969434301862917920422395, 5.12464647385414978552669869920, 5.77083966711572480116835402839, 5.82523467346732724361261848658, 6.55149312304631036741604650053, 6.55186184829324675613080525884, 6.90705818757530641793191366558, 7.53646783731309843545785946743, 7.80337201638930396026904987793, 8.096683657834363950212778636406

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