Properties

Label 4-280e2-1.1-c1e2-0-24
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 2·7-s − 5·9-s + 4·16-s + 6·17-s − 12·23-s + 25-s − 4·28-s − 8·31-s + 10·36-s − 24·41-s + 18·47-s + 3·49-s − 10·63-s − 8·64-s − 12·68-s + 4·73-s − 2·79-s + 16·81-s − 24·89-s + 24·92-s − 2·97-s − 2·100-s + 10·103-s + 8·112-s + 12·113-s + 12·119-s + ⋯
L(s)  = 1  − 4-s + 0.755·7-s − 5/3·9-s + 16-s + 1.45·17-s − 2.50·23-s + 1/5·25-s − 0.755·28-s − 1.43·31-s + 5/3·36-s − 3.74·41-s + 2.62·47-s + 3/7·49-s − 1.25·63-s − 64-s − 1.45·68-s + 0.468·73-s − 0.225·79-s + 16/9·81-s − 2.54·89-s + 2.50·92-s − 0.203·97-s − 1/5·100-s + 0.985·103-s + 0.755·112-s + 1.12·113-s + 1.10·119-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: $\chi_{78400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 78400,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_2$ \( 1 + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−9.462903707627257491442904713279, −8.722556678528362158075028214833, −8.679498853252053444462430241617, −7.980947296320422939728903060714, −7.82866251649990434300481776759, −7.05288221116077551560339282808, −6.15246422175821288222036469083, −5.54934319897223270256186001671, −5.47907564040983295563221596641, −4.78822533627578751004226271654, −3.79583222083421885356258159776, −3.61689003045514298236484607491, −2.59983312100744179860207048134, −1.58268461056007998080671173620, 0, 1.58268461056007998080671173620, 2.59983312100744179860207048134, 3.61689003045514298236484607491, 3.79583222083421885356258159776, 4.78822533627578751004226271654, 5.47907564040983295563221596641, 5.54934319897223270256186001671, 6.15246422175821288222036469083, 7.05288221116077551560339282808, 7.82866251649990434300481776759, 7.980947296320422939728903060714, 8.679498853252053444462430241617, 8.722556678528362158075028214833, 9.462903707627257491442904713279

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