Properties

Label 4-3120e2-1.1-c1e2-0-18
Degree $4$
Conductor $9734400$
Sign $1$
Analytic cond. $620.673$
Root an. cond. $4.99132$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·5-s + 7-s + 3·9-s + 5·11-s − 2·13-s − 4·15-s + 5·17-s + 4·19-s + 2·21-s − 7·23-s + 3·25-s + 4·27-s + 6·29-s + 6·31-s + 10·33-s − 2·35-s − 7·37-s − 4·39-s + 15·41-s − 8·43-s − 6·45-s − 10·47-s − 3·49-s + 10·51-s + 15·53-s − 10·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 0.377·7-s + 9-s + 1.50·11-s − 0.554·13-s − 1.03·15-s + 1.21·17-s + 0.917·19-s + 0.436·21-s − 1.45·23-s + 3/5·25-s + 0.769·27-s + 1.11·29-s + 1.07·31-s + 1.74·33-s − 0.338·35-s − 1.15·37-s − 0.640·39-s + 2.34·41-s − 1.21·43-s − 0.894·45-s − 1.45·47-s − 3/7·49-s + 1.40·51-s + 2.06·53-s − 1.34·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9734400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(620.673\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9734400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.974916833\)
\(L(\frac12)\) \(\approx\) \(4.974916833\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 7 T + 76 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 15 T + 128 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 15 T + 152 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + T + 112 T^{2} + p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - T - 114 T^{2} - p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 3 T + 150 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 3 T + 88 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 17 T + 174 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
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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

−8.601358538160058229187367823700, −8.513012574107738851453777616718, −8.125728187463283125375758241734, −7.921404438105427766392533749462, −7.43836417435185345354999384011, −7.16004176464981220129891423597, −6.59100561894644017456618011091, −6.58306695307714729680760656198, −5.73244919259082633904324530050, −5.52078459065643707212748824791, −4.73006779511847606493815703997, −4.64624501440403093630306086181, −3.91022293250671361286015060771, −3.89791818750877320500180721496, −3.25079129669448397673607378349, −3.03717918648948658688971072550, −2.32221787406482267844107067398, −1.85904083970071039005963747732, −1.16136551100561639049000530401, −0.74182929499742826857730364765, 0.74182929499742826857730364765, 1.16136551100561639049000530401, 1.85904083970071039005963747732, 2.32221787406482267844107067398, 3.03717918648948658688971072550, 3.25079129669448397673607378349, 3.89791818750877320500180721496, 3.91022293250671361286015060771, 4.64624501440403093630306086181, 4.73006779511847606493815703997, 5.52078459065643707212748824791, 5.73244919259082633904324530050, 6.58306695307714729680760656198, 6.59100561894644017456618011091, 7.16004176464981220129891423597, 7.43836417435185345354999384011, 7.921404438105427766392533749462, 8.125728187463283125375758241734, 8.513012574107738851453777616718, 8.601358538160058229187367823700

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