Properties

Label 4-115200-1.1-c1e2-0-11
Degree $4$
Conductor $115200$
Sign $1$
Analytic cond. $7.34525$
Root an. cond. $1.64627$
Motivic weight $1$
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  + 8·7-s + 9-s − 4·17-s − 16·23-s + 25-s + 20·41-s + 16·47-s + 34·49-s + 8·63-s − 28·73-s + 32·79-s + 81-s + 4·89-s + 4·97-s + 8·103-s + 12·113-s − 32·119-s − 22·121-s + ⋯
L(s)  = 1  + 3.02·7-s + 1/3·9-s − 0.970·17-s − 3.33·23-s + 1/5·25-s + 3.12·41-s + 2.33·47-s + 34/7·49-s + 1.00·63-s − 3.27·73-s + 3.60·79-s + 1/9·81-s + 0.423·89-s + 0.406·97-s + 0.788·103-s + 1.12·113-s − 2.93·119-s − 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(115200\)    =    \(2^{9} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(7.34525\)
Root analytic conductor: \(1.64627\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 115200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.191749975\)
\(L(\frac12)\) \(\approx\) \(2.191749975\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.7.ai_be
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.23.q_eg
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.a_di
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.73.bc_ne
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−9.138788409882082860623313179234, −9.082691646086291194533627292052, −8.338450200049268291048738447687, −7.929044973464791963060046214031, −7.64493796343622540106320647343, −7.34594365176835478558310775458, −6.25442273107448540439252397130, −5.92809906508794084871962623137, −5.28897888504504330389281711769, −4.67467612086321652791764027956, −4.13728859524865050162716547657, −4.05477376880357436694530570270, −2.25919867640796924053383456153, −2.19180563382997996757520036314, −1.20192952395638838673519576448, 1.20192952395638838673519576448, 2.19180563382997996757520036314, 2.25919867640796924053383456153, 4.05477376880357436694530570270, 4.13728859524865050162716547657, 4.67467612086321652791764027956, 5.28897888504504330389281711769, 5.92809906508794084871962623137, 6.25442273107448540439252397130, 7.34594365176835478558310775458, 7.64493796343622540106320647343, 7.929044973464791963060046214031, 8.338450200049268291048738447687, 9.082691646086291194533627292052, 9.138788409882082860623313179234

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