Properties

Label 4-160400-1.1-c1e2-0-1
Degree $4$
Conductor $160400$
Sign $-1$
Analytic cond. $10.2272$
Root an. cond. $1.78829$
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 − 3·8-s − 4·9-s − 13-s − 16-s + 9·17-s − 4·18-s + 25-s − 26-s − 4·29-s + 5·32-s + 9·34-s + 4·36-s − 7·37-s − 6·41-s − 13·49-s + 50-s + 52-s + 7·53-s − 4·58-s − 9·61-s + 7·64-s − 9·68-s + 12·72-s − 10·73-s − 7·74-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 4/3·9-s − 0.277·13-s − 1/4·16-s + 2.18·17-s − 0.942·18-s + 1/5·25-s − 0.196·26-s − 0.742·29-s + 0.883·32-s + 1.54·34-s + 2/3·36-s − 1.15·37-s − 0.937·41-s − 1.85·49-s + 0.141·50-s + 0.138·52-s + 0.961·53-s − 0.525·58-s − 1.15·61-s + 7/8·64-s − 1.09·68-s + 1.41·72-s − 1.17·73-s − 0.813·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(160400\)    =    \(2^{4} \cdot 5^{2} \cdot 401\)
Sign: $-1$
Analytic conductor: \(10.2272\)
Root analytic conductor: \(1.78829\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 160400,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
401$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 21 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.11.a_r
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.13.b_u
17$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.17.aj_bq
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.19.a_au
23$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.23.a_bc
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.e_ba
31$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.31.a_e
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.37.h_ce
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.g_k
43$C_2^2$ \( 1 - 57 T^{2} + p^{2} T^{4} \) 2.43.a_acf
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.53.ah_du
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.a_s
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.61.j_fg
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2^2$ \( 1 + 92 T^{2} + p^{2} T^{4} \) 2.71.a_do
73$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.73.k_ff
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.a_abm
83$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.83.a_f
89$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.ba_mk
97$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.ab_hg
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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

−8.940436403926368416335594559340, −8.519676752647461614256283092940, −8.106349007601805700443549294575, −7.63570769649469769781178379238, −7.00072811522805407743764715947, −6.34664111409846251353130556489, −5.78907975307412042850252441597, −5.39209687096862046210434606764, −5.14556423289996085871398384573, −4.37368544580895914221986171533, −3.59689423786470358298159733391, −3.22185264937223716010891835646, −2.71767561138584126858003350655, −1.47080792365288627533526972929, 0, 1.47080792365288627533526972929, 2.71767561138584126858003350655, 3.22185264937223716010891835646, 3.59689423786470358298159733391, 4.37368544580895914221986171533, 5.14556423289996085871398384573, 5.39209687096862046210434606764, 5.78907975307412042850252441597, 6.34664111409846251353130556489, 7.00072811522805407743764715947, 7.63570769649469769781178379238, 8.106349007601805700443549294575, 8.519676752647461614256283092940, 8.940436403926368416335594559340

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