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Properties

Label 4-280e2-1.1-c1e2-0-1
Degree $4$
Conductor $78400$
Sign $1$
Analytic cond. $4.99885$
Root an. cond. $1.49526$
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  − 4·5-s − 2·9-s − 4·19-s + 11·25-s + 4·29-s + 8·31-s − 4·41-s + 8·45-s + 49-s + 12·59-s + 8·61-s − 16·79-s − 5·81-s + 20·89-s + 16·95-s + 24·101-s + 20·109-s − 22·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s − 32·155-s + ⋯
L(s)  = 1  − 1.78·5-s − 2/3·9-s − 0.917·19-s + 11/5·25-s + 0.742·29-s + 1.43·31-s − 0.624·41-s + 1.19·45-s + 1/7·49-s + 1.56·59-s + 1.02·61-s − 1.80·79-s − 5/9·81-s + 2.11·89-s + 1.64·95-s + 2.38·101-s + 1.91·109-s − 2·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(78400\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4.99885\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 78400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8038577114\)
\(L(\frac12)\) \(\approx\) \(0.8038577114\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.a_da
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.61.ai_fi
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.a_aby
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$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−9.962370660926456280013102621248, −8.928329697232844766585907221767, −8.694697440739683427951548718991, −8.188057214509100026281306554311, −7.965971686397333220186526353962, −7.14438434722230698367292292955, −6.86591116390976664813936408049, −6.19535765189485358396007496237, −5.55399016745426097333734158356, −4.61889361197427675528739680931, −4.50185552371175935345657331179, −3.62433266137155427382097354955, −3.14137792992390816749783057891, −2.29482126376589182159266782001, −0.68915328699946070952810996679, 0.68915328699946070952810996679, 2.29482126376589182159266782001, 3.14137792992390816749783057891, 3.62433266137155427382097354955, 4.50185552371175935345657331179, 4.61889361197427675528739680931, 5.55399016745426097333734158356, 6.19535765189485358396007496237, 6.86591116390976664813936408049, 7.14438434722230698367292292955, 7.965971686397333220186526353962, 8.188057214509100026281306554311, 8.694697440739683427951548718991, 8.928329697232844766585907221767, 9.962370660926456280013102621248

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