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Properties

Label 4-240e2-1.1-c1e2-0-10
Degree $4$
Conductor $57600$
Sign $1$
Analytic cond. $3.67262$
Root an. cond. $1.38434$
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 − 4·7-s + 9-s + 4·13-s + 8·19-s − 8·21-s + 25-s − 4·27-s + 8·31-s + 4·37-s + 8·39-s + 20·43-s − 2·49-s + 16·57-s + 4·61-s − 4·63-s − 4·67-s + 4·73-s + 2·75-s − 16·79-s − 11·81-s − 16·91-s + 16·93-s + 4·97-s − 28·103-s + 4·109-s + 8·111-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.51·7-s + 1/3·9-s + 1.10·13-s + 1.83·19-s − 1.74·21-s + 1/5·25-s − 0.769·27-s + 1.43·31-s + 0.657·37-s + 1.28·39-s + 3.04·43-s − 2/7·49-s + 2.11·57-s + 0.512·61-s − 0.503·63-s − 0.488·67-s + 0.468·73-s + 0.230·75-s − 1.80·79-s − 1.22·81-s − 1.67·91-s + 1.65·93-s + 0.406·97-s − 2.75·103-s + 0.383·109-s + 0.759·111-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 57600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.854188003\)
\(L(\frac12)\) \(\approx\) \(1.854188003\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
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.au_he
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
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.e_fi
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−9.683565298540966130897522415203, −9.633261704500415779063377753491, −8.847850276232423443211930888771, −8.740293324354028093320034492248, −7.74818737522785607031159368119, −7.71273110823499542846308181544, −6.88656092970954199609923322632, −6.27087624192875571051851265258, −5.90138678053477774939394585997, −5.17606419420266207753384183426, −4.12250433368686324236236368171, −3.70610300459195515417889608627, −2.84705552122166969163585152299, −2.76929890617261215013507568311, −1.16915866227454488376692012454, 1.16915866227454488376692012454, 2.76929890617261215013507568311, 2.84705552122166969163585152299, 3.70610300459195515417889608627, 4.12250433368686324236236368171, 5.17606419420266207753384183426, 5.90138678053477774939394585997, 6.27087624192875571051851265258, 6.88656092970954199609923322632, 7.71273110823499542846308181544, 7.74818737522785607031159368119, 8.740293324354028093320034492248, 8.847850276232423443211930888771, 9.633261704500415779063377753491, 9.683565298540966130897522415203

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