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Properties

Label 4-260e2-1.1-c0e2-0-3
Degree $4$
Conductor $67600$
Sign $1$
Analytic cond. $0.0168368$
Root an. cond. $0.360217$
Motivic weight $0$
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-s + 2·5-s − 2·13-s + 16-s − 2·17-s − 2·20-s + 3·25-s − 2·41-s + 2·49-s + 2·52-s − 2·53-s − 64-s − 4·65-s + 2·68-s + 2·80-s − 81-s − 4·85-s − 2·89-s − 3·100-s + 2·109-s + 2·113-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  − 4-s + 2·5-s − 2·13-s + 16-s − 2·17-s − 2·20-s + 3·25-s − 2·41-s + 2·49-s + 2·52-s − 2·53-s − 64-s − 4·65-s + 2·68-s + 2·80-s − 81-s − 4·85-s − 2·89-s − 3·100-s + 2·109-s + 2·113-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(67600\)    =    \(2^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.0168368\)
Root analytic conductor: \(0.360217\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 67600,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5669098056\)
\(L(\frac12)\) \(\approx\) \(0.5669098056\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−12.82251058420569389269467634263, −12.23150478996913530495027146426, −11.45088335021371212431244632391, −10.92493001509926890157433226094, −10.21572661603365946786259719149, −10.11698450147350455664565324277, −9.558298903833028026275004216819, −9.269196707780356329923341568476, −8.674766221919918008550416064948, −8.446478796280666770845829155343, −7.41235402654655989524850721661, −6.97217871667529371634475740297, −6.46760654973349580082843120313, −5.77247295088478574941748985570, −5.32889621241390485713710346214, −4.60645839685577687682105807730, −4.56557210181478800039033333937, −3.19491018413712777771327149791, −2.43061537449341308707134139626, −1.81103030117855911204439404581, 1.81103030117855911204439404581, 2.43061537449341308707134139626, 3.19491018413712777771327149791, 4.56557210181478800039033333937, 4.60645839685577687682105807730, 5.32889621241390485713710346214, 5.77247295088478574941748985570, 6.46760654973349580082843120313, 6.97217871667529371634475740297, 7.41235402654655989524850721661, 8.446478796280666770845829155343, 8.674766221919918008550416064948, 9.269196707780356329923341568476, 9.558298903833028026275004216819, 10.11698450147350455664565324277, 10.21572661603365946786259719149, 10.92493001509926890157433226094, 11.45088335021371212431244632391, 12.23150478996913530495027146426, 12.82251058420569389269467634263

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