Properties

Label 4-280e2-1.1-c1e2-0-29
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  + 2·3-s − 2·4-s − 3·9-s − 6·11-s − 4·12-s + 4·16-s + 6·17-s + 4·19-s + 25-s − 14·27-s − 12·33-s + 6·36-s − 24·41-s − 20·43-s + 12·44-s + 8·48-s + 49-s + 12·51-s + 8·57-s − 8·64-s − 8·67-s − 12·68-s + 4·73-s + 2·75-s − 8·76-s − 4·81-s + 24·83-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s − 9-s − 1.80·11-s − 1.15·12-s + 16-s + 1.45·17-s + 0.917·19-s + 1/5·25-s − 2.69·27-s − 2.08·33-s + 36-s − 3.74·41-s − 3.04·43-s + 1.80·44-s + 1.15·48-s + 1/7·49-s + 1.68·51-s + 1.05·57-s − 64-s − 0.977·67-s − 1.45·68-s + 0.468·73-s + 0.230·75-s − 0.917·76-s − 4/9·81-s + 2.63·83-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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
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

−9.581459961422691318510933413886, −8.679498853252053444462430241617, −8.611548044991135538391646913928, −7.980947296320422939728903060714, −7.937110667039146160138689071489, −7.21068288194212183905414731057, −6.34730906119503780706199760723, −5.47907564040983295563221596641, −5.25086715638915413911306599520, −4.93400384358317586060280833112, −3.61689003045514298236484607491, −3.24103653131434423009222437756, −2.94130668011304353115387148308, −1.79896277633481319941660230821, 0, 1.79896277633481319941660230821, 2.94130668011304353115387148308, 3.24103653131434423009222437756, 3.61689003045514298236484607491, 4.93400384358317586060280833112, 5.25086715638915413911306599520, 5.47907564040983295563221596641, 6.34730906119503780706199760723, 7.21068288194212183905414731057, 7.937110667039146160138689071489, 7.980947296320422939728903060714, 8.611548044991135538391646913928, 8.679498853252053444462430241617, 9.581459961422691318510933413886

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