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Properties

Label 4-1040e2-1.1-c1e2-0-22
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  + 9-s − 5·13-s + 3·17-s + 25-s + 15·29-s + 6·37-s − 6·41-s + 4·49-s + 13·61-s + 9·73-s − 8·81-s − 12·89-s − 12·97-s + 15·101-s + 3·109-s + 21·113-s − 5·117-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 3·153-s + 157-s + 163-s + ⋯
L(s)  = 1  + 1/3·9-s − 1.38·13-s + 0.727·17-s + 1/5·25-s + 2.78·29-s + 0.986·37-s − 0.937·41-s + 4/7·49-s + 1.66·61-s + 1.05·73-s − 8/9·81-s − 1.27·89-s − 1.21·97-s + 1.49·101-s + 0.287·109-s + 1.97·113-s − 0.462·117-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.242·153-s + 0.0798·157-s + 0.0783·163-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

\(L(1)\) \(\approx\) \(2.029631328\)
\(L(\frac12)\) \(\approx\) \(2.029631328\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
11$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.11.a_i
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.ad_q
19$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.19.a_aq
23$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.23.a_au
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.29.ap_ei
31$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.31.a_f
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.ag_bi
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.g_de
43$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.43.a_abm
47$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.47.a_ae
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.59.a_abc
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.61.an_gg
67$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \) 2.67.a_acj
71$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.71.a_l
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.aj_eu
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2^2$ \( 1 + 137 T^{2} + p^{2} T^{4} \) 2.83.a_fh
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.m_gw
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.m_ig
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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

−8.076186530294258249926119986844, −7.61861064042466201857160481167, −7.17883079504766284955093929049, −6.85285439451779141597251235564, −6.36127115765269371318949736574, −5.93096764086199705885714992410, −5.23338769860101578507479850551, −4.95422272607673637440994285119, −4.53170706915980558079217404515, −3.97396620341470168723768770026, −3.35333174366830349130200833166, −2.65031657285860588784292355290, −2.44891333077042494832889662763, −1.42973323474069566478633011569, −0.68735955798413477108153608942, 0.68735955798413477108153608942, 1.42973323474069566478633011569, 2.44891333077042494832889662763, 2.65031657285860588784292355290, 3.35333174366830349130200833166, 3.97396620341470168723768770026, 4.53170706915980558079217404515, 4.95422272607673637440994285119, 5.23338769860101578507479850551, 5.93096764086199705885714992410, 6.36127115765269371318949736574, 6.85285439451779141597251235564, 7.17883079504766284955093929049, 7.61861064042466201857160481167, 8.076186530294258249926119986844

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