Properties

Label 4-1040e2-1.1-c1e2-0-11
Degree $4$
Conductor $1081600$
Sign $1$
Analytic cond. $68.9637$
Root an. cond. $2.88174$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 4·9-s − 2·13-s + 6·17-s + 4·25-s − 3·29-s − 18·37-s − 6·41-s − 12·45-s − 2·49-s + 19·61-s + 6·65-s + 12·73-s + 7·81-s − 18·85-s + 18·97-s − 15·101-s − 9·109-s + 3·113-s − 8·117-s − 2·121-s + 3·125-s + ⋯
L(s)  = 1  − 1.34·5-s + 4/3·9-s − 0.554·13-s + 1.45·17-s + 4/5·25-s − 0.557·29-s − 2.95·37-s − 0.937·41-s − 1.78·45-s − 2/7·49-s + 2.43·61-s + 0.744·65-s + 1.40·73-s + 7/9·81-s − 1.95·85-s + 1.82·97-s − 1.49·101-s − 0.862·109-s + 0.282·113-s − 0.739·117-s − 0.181·121-s + 0.268·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1081600\)    =    \(2^{8} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(68.9637\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1081600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.392478546\)
\(L(\frac12)\) \(\approx\) \(1.392478546\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + 3 T + p T^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.11.a_c
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ag_bi
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.23.a_af
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.d_cg
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.31.a_ak
37$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.s_fy
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.41.g_dn
43$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.43.a_au
47$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.47.a_x
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.59.a_abc
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.61.at_hk
67$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \) 2.67.a_ea
71$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \) 2.71.a_acd
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.73.am_gk
79$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.79.a_fd
83$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.83.a_c
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.97.as_jq
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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

−7.992711265733549012877608591748, −7.66118157760456704815322554787, −7.16638933071835913862567322391, −6.93008978638425165144508807246, −6.61816405573370011843086202708, −5.74486844488638259496920440257, −5.18355195565442058493642595443, −5.05107683558691116459023756977, −4.33680901692238710598412392787, −3.78228306817423463112733042265, −3.59650197606838722334130414641, −3.05233013887686801847767884059, −2.07162894594675148115507169565, −1.50762052733478522008932948794, −0.55970589121774674667477117002, 0.55970589121774674667477117002, 1.50762052733478522008932948794, 2.07162894594675148115507169565, 3.05233013887686801847767884059, 3.59650197606838722334130414641, 3.78228306817423463112733042265, 4.33680901692238710598412392787, 5.05107683558691116459023756977, 5.18355195565442058493642595443, 5.74486844488638259496920440257, 6.61816405573370011843086202708, 6.93008978638425165144508807246, 7.16638933071835913862567322391, 7.66118157760456704815322554787, 7.992711265733549012877608591748

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