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Properties

Label 4-60e3-1.1-c1e2-0-19
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  − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s − 2·9-s − 2·10-s + 2·12-s − 3·13-s + 15-s − 4·16-s + 4·18-s + 2·20-s + 25-s + 6·26-s − 5·27-s − 2·30-s − 2·31-s + 8·32-s − 4·36-s + 12·37-s − 3·39-s − 3·41-s − 3·43-s − 2·45-s − 4·48-s − 10·49-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s − 2/3·9-s − 0.632·10-s + 0.577·12-s − 0.832·13-s + 0.258·15-s − 16-s + 0.942·18-s + 0.447·20-s + 1/5·25-s + 1.17·26-s − 0.962·27-s − 0.365·30-s − 0.359·31-s + 1.41·32-s − 2/3·36-s + 1.97·37-s − 0.480·39-s − 0.468·41-s − 0.457·43-s − 0.298·45-s − 0.577·48-s − 1.42·49-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$C_2$ \( 1 + p T + p T^{2} \)
3$C_2$ \( 1 - T + p T^{2} \)
5$C_1$ \( 1 - T \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \) 2.11.a_am
13$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.d_i
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.19.a_an
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \) 2.23.a_av
29$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.29.a_abg
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.c_cc
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.am_dq
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.41.d_cu
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.43.d_dk
47$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \) 2.47.a_ap
53$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.53.ah_i
59$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \) 2.59.a_adm
61$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \) 2.61.a_dr
67$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.j_du
71$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.71.j_cu
73$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.73.a_dw
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.k_dy
83$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.s_je
89$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.89.v_lc
97$C_2^2$ \( 1 - 176 T^{2} + p^{2} T^{4} \) 2.97.a_agu
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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

−8.719478234935615867978748867412, −8.478896065053912585392389292237, −8.060112442021522775444254290557, −7.37098860384523408563181956512, −7.28707590353294566879593311442, −6.56325927203695683308289102646, −5.92677060926460013974694426257, −5.55403496446035493943234474244, −4.67832390952108040223362382520, −4.33711019032836985145843407755, −3.30018857754011546651837966620, −2.72648121336042217986362913729, −2.15117827428324848537758894218, −1.34441859288597537687459084223, 0, 1.34441859288597537687459084223, 2.15117827428324848537758894218, 2.72648121336042217986362913729, 3.30018857754011546651837966620, 4.33711019032836985145843407755, 4.67832390952108040223362382520, 5.55403496446035493943234474244, 5.92677060926460013974694426257, 6.56325927203695683308289102646, 7.28707590353294566879593311442, 7.37098860384523408563181956512, 8.060112442021522775444254290557, 8.478896065053912585392389292237, 8.719478234935615867978748867412

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