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Properties

Label 4-85280-1.1-c1e2-0-2
Degree $4$
Conductor $85280$
Sign $-1$
Analytic cond. $5.43752$
Root an. cond. $1.52703$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 8-s − 2·9-s + 10-s + 3·13-s + 16-s − 6·17-s + 2·18-s − 20-s − 4·25-s − 3·26-s − 32-s + 6·34-s − 2·36-s − 2·37-s + 40-s + 7·41-s + 2·45-s − 4·49-s + 4·50-s + 3·52-s − 6·53-s − 8·61-s + 64-s − 3·65-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.832·13-s + 1/4·16-s − 1.45·17-s + 0.471·18-s − 0.223·20-s − 4/5·25-s − 0.588·26-s − 0.176·32-s + 1.02·34-s − 1/3·36-s − 0.328·37-s + 0.158·40-s + 1.09·41-s + 0.298·45-s − 4/7·49-s + 0.565·50-s + 0.416·52-s − 0.824·53-s − 1.02·61-s + 1/8·64-s − 0.372·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85280 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85280 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(85280\)    =    \(2^{5} \cdot 5 \cdot 13 \cdot 41\)
Sign: $-1$
Analytic conductor: \(5.43752\)
Root analytic conductor: \(1.52703\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 85280,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 6 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.11.a_ai
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.g_bi
19$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.19.a_ai
23$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.23.a_ao
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.31.a_w
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.c_co
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.47.a_bo
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.g_bi
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.59.a_bu
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.67.a_dw
71$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.71.a_cy
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.o_ek
79$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.79.a_adi
83$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \) 2.83.a_afe
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.m_cs
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.97.aw_lu
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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

−9.246941057515520090121407704629, −8.988112755763923473567949959109, −8.443168764159549754361617318408, −8.093495499077584418144224079178, −7.49645022137580213864703164208, −7.02838848597100029296855351179, −6.25486874882998498625565081776, −6.09579540538470638332786209985, −5.32136951849989398797907059640, −4.50401699733099909432370760322, −3.97718011523340342498385759708, −3.18342792008020979736026817680, −2.47890192988795359157034902710, −1.52552087378986494284403548026, 0, 1.52552087378986494284403548026, 2.47890192988795359157034902710, 3.18342792008020979736026817680, 3.97718011523340342498385759708, 4.50401699733099909432370760322, 5.32136951849989398797907059640, 6.09579540538470638332786209985, 6.25486874882998498625565081776, 7.02838848597100029296855351179, 7.49645022137580213864703164208, 8.093495499077584418144224079178, 8.443168764159549754361617318408, 8.988112755763923473567949959109, 9.246941057515520090121407704629

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