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Properties

Label 4-240e2-1.1-c1e2-0-9
Degree $4$
Conductor $57600$
Sign $1$
Analytic cond. $3.67262$
Root an. cond. $1.38434$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 9-s − 25-s + 12·29-s + 12·41-s − 2·45-s + 14·49-s + 4·61-s + 81-s − 12·89-s − 36·101-s + 4·109-s − 6·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 0.894·5-s − 1/3·9-s − 1/5·25-s + 2.22·29-s + 1.87·41-s − 0.298·45-s + 2·49-s + 0.512·61-s + 1/9·81-s − 1.27·89-s − 3.58·101-s + 0.383·109-s − 0.545·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(57600\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(3.67262\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 57600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.625776610\)
\(L(\frac12)\) \(\approx\) \(1.625776610\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.97.a_ak
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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

−10.01310549089899510498567713509, −9.516767031653194228293505000223, −8.999222769477543579645116967896, −8.532532851007177915373363419493, −8.009712187101956172091092596951, −7.39696023102605292005128379490, −6.75626336855671302421092609277, −6.27483845369351885835071661051, −5.70601464750226217237219593757, −5.28260549942834099623543513745, −4.45715443970348700585757553785, −3.90964196321524344299090893356, −2.79608921257616715227246506752, −2.40914995981068008590123379595, −1.16055652003619941740790198107, 1.16055652003619941740790198107, 2.40914995981068008590123379595, 2.79608921257616715227246506752, 3.90964196321524344299090893356, 4.45715443970348700585757553785, 5.28260549942834099623543513745, 5.70601464750226217237219593757, 6.27483845369351885835071661051, 6.75626336855671302421092609277, 7.39696023102605292005128379490, 8.009712187101956172091092596951, 8.532532851007177915373363419493, 8.999222769477543579645116967896, 9.516767031653194228293505000223, 10.01310549089899510498567713509

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