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Properties

Label 4-60e3-1.1-c1e2-0-18
Degree $4$
Conductor $216000$
Sign $-1$
Analytic cond. $13.7723$
Root an. cond. $1.92642$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s + 5-s + 9-s + 2·12-s + 2·13-s − 15-s + 4·16-s − 2·20-s + 25-s − 27-s − 8·31-s − 2·36-s − 10·37-s − 2·39-s − 18·41-s − 4·43-s + 45-s − 4·48-s + 2·49-s − 4·52-s + 12·53-s + 2·60-s − 8·64-s + 2·65-s + 8·67-s − 75-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.447·5-s + 1/3·9-s + 0.577·12-s + 0.554·13-s − 0.258·15-s + 16-s − 0.447·20-s + 1/5·25-s − 0.192·27-s − 1.43·31-s − 1/3·36-s − 1.64·37-s − 0.320·39-s − 2.81·41-s − 0.609·43-s + 0.149·45-s − 0.577·48-s + 2/7·49-s − 0.554·52-s + 1.64·53-s + 0.258·60-s − 64-s + 0.248·65-s + 0.977·67-s − 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(216000\)    =    \(2^{6} \cdot 3^{3} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(13.7723\)
Root analytic conductor: \(1.92642\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 216000,\ (\ :1/2, 1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 + p T^{2} \)
3$C_1$ \( 1 + T \)
5$C_1$ \( 1 - T \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.ac_s
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.23.a_ao
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.k_dm
41$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.s_fy
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.e_cc
47$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.47.a_aba
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.59.a_adu
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.73.a_acw
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.i_be
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ag_gw
97$C_2^2$ \( 1 - 170 T^{2} + p^{2} T^{4} \) 2.97.a_ago
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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

−8.724106438367421303345162639094, −8.542414096846067177198244976201, −7.982913945729478238750702527615, −7.23853334997921135605777421359, −6.88247170613678561810180280311, −6.37477083063926939926298893868, −5.67797429529676722720135763007, −5.24149311167588692091359783047, −5.09255145585540175993384578396, −4.21040799738456801763099304038, −3.68350046726164799970870121574, −3.22951288219260696620707097305, −2.04071424181164247240988898380, −1.31631268356932863582550653017, 0, 1.31631268356932863582550653017, 2.04071424181164247240988898380, 3.22951288219260696620707097305, 3.68350046726164799970870121574, 4.21040799738456801763099304038, 5.09255145585540175993384578396, 5.24149311167588692091359783047, 5.67797429529676722720135763007, 6.37477083063926939926298893868, 6.88247170613678561810180280311, 7.23853334997921135605777421359, 7.982913945729478238750702527615, 8.542414096846067177198244976201, 8.724106438367421303345162639094

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