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Properties

Label 4-85280-1.1-c1e2-0-3
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 + 2·5-s − 8-s − 4·9-s − 2·10-s + 16-s − 12·17-s + 4·18-s + 2·20-s + 2·25-s − 29-s − 32-s + 12·34-s − 4·36-s + 6·37-s − 2·40-s − 9·41-s − 8·45-s + 12·49-s − 2·50-s − 13·53-s + 58-s − 3·61-s + 64-s − 12·68-s + 4·72-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 4/3·9-s − 0.632·10-s + 1/4·16-s − 2.91·17-s + 0.942·18-s + 0.447·20-s + 2/5·25-s − 0.185·29-s − 0.176·32-s + 2.05·34-s − 2/3·36-s + 0.986·37-s − 0.316·40-s − 1.40·41-s − 1.19·45-s + 12/7·49-s − 0.282·50-s − 1.78·53-s + 0.131·58-s − 0.384·61-s + 1/8·64-s − 1.45·68-s + 0.471·72-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$ \( ( 1 + T )( 1 - 3 T + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 8 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \) 2.7.a_am
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
17$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.m_cr
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.23.a_bo
29$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.b_ce
31$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \) 2.31.a_bk
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ag_cg
43$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.43.a_abq
47$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.47.a_ca
53$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.n_fg
59$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \) 2.59.a_bd
61$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.d_eu
67$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.67.a_l
71$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.71.a_ade
73$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.e_fq
79$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.79.a_q
83$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.83.a_bq
89$C_2$$\times$$C_2$ \( ( 1 + 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.ba_ni
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.97.d_cm
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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

−9.224486339207784274856831695517, −9.014324528663914988415195950125, −8.598176117075040107317017705902, −8.167669062704529546265040923924, −7.44719866346887288604506445660, −6.75510580641326044100052274163, −6.46024845359945043489505309416, −5.93744053383392269601718150667, −5.38053924691955567999028957967, −4.68415335700728420746058609766, −4.00764230900292277926282448475, −2.89503764334687303865924893224, −2.47423862222541547451291809286, −1.71510783331884397824148465439, 0, 1.71510783331884397824148465439, 2.47423862222541547451291809286, 2.89503764334687303865924893224, 4.00764230900292277926282448475, 4.68415335700728420746058609766, 5.38053924691955567999028957967, 5.93744053383392269601718150667, 6.46024845359945043489505309416, 6.75510580641326044100052274163, 7.44719866346887288604506445660, 8.167669062704529546265040923924, 8.598176117075040107317017705902, 9.014324528663914988415195950125, 9.224486339207784274856831695517

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