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Properties

Label 4-72000-1.1-c1e2-0-8
Degree $4$
Conductor $72000$
Sign $1$
Analytic cond. $4.59078$
Root an. cond. $1.46376$
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 + 5-s + 4·7-s + 9-s − 2·15-s + 12·17-s − 8·21-s + 25-s + 4·27-s + 4·35-s − 20·43-s + 45-s − 2·49-s − 24·51-s + 12·53-s − 24·59-s + 4·61-s + 4·63-s + 4·67-s + 24·71-s − 2·75-s − 11·81-s + 12·85-s + 28·103-s − 8·105-s + 4·109-s + 12·113-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.516·15-s + 2.91·17-s − 1.74·21-s + 1/5·25-s + 0.769·27-s + 0.676·35-s − 3.04·43-s + 0.149·45-s − 2/7·49-s − 3.36·51-s + 1.64·53-s − 3.12·59-s + 0.512·61-s + 0.503·63-s + 0.488·67-s + 2.84·71-s − 0.230·75-s − 1.22·81-s + 1.30·85-s + 2.75·103-s − 0.780·105-s + 0.383·109-s + 1.12·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(72000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(4.59078\)
Root analytic conductor: \(1.46376\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 72000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.335360080\)
\(L(\frac12)\) \(\approx\) \(1.335360080\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
5$C_1$ \( 1 - T \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.43.u_he
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−9.749524176059453166876684221310, −9.703656252458157439228300963800, −8.536040373596851277524077678909, −8.407676785947288187996164912905, −7.64589778281564359800589548721, −7.47121424818625422471458115292, −6.43307814142801241380910059979, −6.21444115782927123192608967686, −5.30602566274475700528497955431, −5.19647848534431193907436500640, −4.82118726362395324834121081499, −3.71119793411816721603957340078, −3.07875775618643284846845606551, −1.81793015252092636076156145980, −1.10363850080882238424508077779, 1.10363850080882238424508077779, 1.81793015252092636076156145980, 3.07875775618643284846845606551, 3.71119793411816721603957340078, 4.82118726362395324834121081499, 5.19647848534431193907436500640, 5.30602566274475700528497955431, 6.21444115782927123192608967686, 6.43307814142801241380910059979, 7.47121424818625422471458115292, 7.64589778281564359800589548721, 8.407676785947288187996164912905, 8.536040373596851277524077678909, 9.703656252458157439228300963800, 9.749524176059453166876684221310

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