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Properties

Label 4-240e2-1.1-c1e2-0-32
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 9-s − 25-s − 8·31-s − 12·41-s − 8·47-s + 2·49-s + 4·63-s − 16·71-s + 20·73-s − 8·79-s + 81-s + 12·89-s + 4·97-s − 20·103-s − 8·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯
L(s)  = 1  − 1.51·7-s − 1/3·9-s − 1/5·25-s − 1.43·31-s − 1.87·41-s − 1.16·47-s + 2/7·49-s + 0.503·63-s − 1.89·71-s + 2.34·73-s − 0.900·79-s + 1/9·81-s + 1.27·89-s + 0.406·97-s − 1.97·103-s − 0.752·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-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: \(1\)
Selberg data: \((4,\ 57600,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + T^{2} \)
good7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.e_o
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.i_ck
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.a_cg
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.m_dy
43$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.43.a_acc
47$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.i_ec
53$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.53.a_adi
59$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.59.a_bq
61$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.61.a_abq
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.q_hi
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.73.au_iw
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.i_gc
83$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.83.a_adi
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.89.am_hq
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.ae_ha
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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

−9.731165254200104943256288475859, −9.305105979499750932454228317401, −8.809826645876322506737955261902, −8.245731709282772253188317593201, −7.66923514874669183620554411553, −6.99633859425930818805324226984, −6.58205012953080931163933328415, −6.13077306062514132451431095306, −5.43486074061117596459630394809, −4.93087965132095068268358123213, −3.94091703780078230463130708591, −3.41498290649813059166192679797, −2.85634942869738531000039787839, −1.77764217441541021456177625268, 0, 1.77764217441541021456177625268, 2.85634942869738531000039787839, 3.41498290649813059166192679797, 3.94091703780078230463130708591, 4.93087965132095068268358123213, 5.43486074061117596459630394809, 6.13077306062514132451431095306, 6.58205012953080931163933328415, 6.99633859425930818805324226984, 7.66923514874669183620554411553, 8.245731709282772253188317593201, 8.809826645876322506737955261902, 9.305105979499750932454228317401, 9.731165254200104943256288475859

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