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Properties

Label 4-260e2-1.1-c0e2-0-0
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 2·17-s − 25-s − 6·32-s − 4·34-s − 2·41-s − 2·49-s + 2·50-s − 2·53-s + 7·64-s + 6·68-s + 4·73-s − 81-s + 4·82-s + 2·89-s + 4·98-s − 3·100-s + 4·106-s − 2·109-s + 2·113-s + 127-s − 8·128-s + 131-s − 8·136-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 2·17-s − 25-s − 6·32-s − 4·34-s − 2·41-s − 2·49-s + 2·50-s − 2·53-s + 7·64-s + 6·68-s + 4·73-s − 81-s + 4·82-s + 2·89-s + 4·98-s − 3·100-s + 4·106-s − 2·109-s + 2·113-s + 127-s − 8·128-s + 131-s − 8·136-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

\(L(\frac{1}{2})\) \(\approx\) \(0.2559476529\)
\(L(\frac12)\) \(\approx\) \(0.2559476529\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$ \( ( 1 - T )^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−12.03868078674636152471222880535, −12.02582929086622292141981981243, −11.20980221048105044725047029261, −11.04239992511272870701939745140, −10.37317329210454955630126998659, −9.815536100597621499556067821481, −9.749092543078072203135029193729, −9.274106424248876804879528439435, −8.478296824641379647754730030440, −8.109643467058337085922005046852, −7.81045689213456306064817317062, −7.32346580397877620284854966713, −6.53443278402019860554475465853, −6.32641308821268774433648266607, −5.53657516302876886290621075646, −5.00857670055161515029135103719, −3.44649809219711510079715086898, −3.33858863379832676236069890591, −2.16114574364275784524204190894, −1.37736054286372667276825657430, 1.37736054286372667276825657430, 2.16114574364275784524204190894, 3.33858863379832676236069890591, 3.44649809219711510079715086898, 5.00857670055161515029135103719, 5.53657516302876886290621075646, 6.32641308821268774433648266607, 6.53443278402019860554475465853, 7.32346580397877620284854966713, 7.81045689213456306064817317062, 8.109643467058337085922005046852, 8.478296824641379647754730030440, 9.274106424248876804879528439435, 9.749092543078072203135029193729, 9.815536100597621499556067821481, 10.37317329210454955630126998659, 11.04239992511272870701939745140, 11.20980221048105044725047029261, 12.02582929086622292141981981243, 12.03868078674636152471222880535

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