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Properties

Label 4-85280-1.1-c1e2-0-0
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 − 5·5-s − 8-s + 5·10-s + 5·13-s + 16-s − 2·17-s − 5·20-s + 16·25-s − 5·26-s − 10·29-s − 32-s + 2·34-s + 5·40-s − 41-s − 2·49-s − 16·50-s + 5·52-s + 4·53-s + 10·58-s + 14·61-s + 64-s − 25·65-s − 2·68-s − 12·73-s − 5·80-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 2.23·5-s − 0.353·8-s + 1.58·10-s + 1.38·13-s + 1/4·16-s − 0.485·17-s − 1.11·20-s + 16/5·25-s − 0.980·26-s − 1.85·29-s − 0.176·32-s + 0.342·34-s + 0.790·40-s − 0.156·41-s − 2/7·49-s − 2.26·50-s + 0.693·52-s + 0.549·53-s + 1.31·58-s + 1.79·61-s + 1/8·64-s − 3.10·65-s − 0.242·68-s − 1.40·73-s − 0.559·80-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$ \( ( 1 + T )( 1 + 4 T + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 4 T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.11.a_q
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.c_ba
19$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.19.a_m
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
29$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.k_de
31$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.31.a_w
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
43$C_2^2$ \( 1 - 36 T^{2} + p^{2} T^{4} \) 2.43.a_abk
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.ae_dq
59$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.59.a_bo
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.61.ao_go
67$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.67.a_abu
71$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.71.a_cg
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.m_gk
79$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.79.a_cw
83$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.83.a_c
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ac_gw
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.97.e_c
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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

−9.271662652409530632452278918087, −8.702462810800315837806570747685, −8.520432278449695161035005562201, −7.994910072950875533146183239434, −7.49463150062587607101443361388, −7.13802647449479708184660999226, −6.59407133889208597408219882900, −5.91654362493742085403982810019, −5.20763117932080415917962836803, −4.39560427617215298319464361940, −3.73030783611233631745529122298, −3.60498741756709191300900040033, −2.57413862136445485631009545164, −1.28457922168689811575601370703, 0, 1.28457922168689811575601370703, 2.57413862136445485631009545164, 3.60498741756709191300900040033, 3.73030783611233631745529122298, 4.39560427617215298319464361940, 5.20763117932080415917962836803, 5.91654362493742085403982810019, 6.59407133889208597408219882900, 7.13802647449479708184660999226, 7.49463150062587607101443361388, 7.994910072950875533146183239434, 8.520432278449695161035005562201, 8.702462810800315837806570747685, 9.271662652409530632452278918087

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