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Properties

Label 4-80e2-1.1-c1e2-0-10
Degree $4$
Conductor $6400$
Sign $1$
Analytic cond. $0.408069$
Root an. cond. $0.799251$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 2·5-s − 6·7-s + 2·9-s + 4·10-s − 2·11-s − 4·13-s − 12·14-s − 4·16-s + 2·17-s + 4·18-s + 6·19-s + 4·20-s − 4·22-s − 2·23-s − 25-s − 8·26-s − 12·28-s + 14·29-s − 8·32-s + 4·34-s − 12·35-s + 4·36-s − 12·37-s + 12·38-s + 8·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.894·5-s − 2.26·7-s + 2/3·9-s + 1.26·10-s − 0.603·11-s − 1.10·13-s − 3.20·14-s − 16-s + 0.485·17-s + 0.942·18-s + 1.37·19-s + 0.894·20-s − 0.852·22-s − 0.417·23-s − 1/5·25-s − 1.56·26-s − 2.26·28-s + 2.59·29-s − 1.41·32-s + 0.685·34-s − 2.02·35-s + 2/3·36-s − 1.97·37-s + 1.94·38-s + 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.408069\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.575788268\)
\(L(\frac12)\) \(\approx\) \(1.575788268\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.7.g_s
11$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_c
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_c
19$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_s
23$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_c
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.29.ao_du
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.31.a_acg
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.41.a_aco
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.47.ao_du
53$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.53.a_abq
59$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_s
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.c_c
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_s
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \) 2.83.a_agg
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.97.w_ji
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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

−14.15112524404569176674908270993, −14.13143550792508788546022118053, −13.56263174322409099297897186637, −13.18539645043385362545486359499, −12.49276997095376938063047430271, −12.18801456944399837843399013207, −12.06861790428375561600779554210, −10.66720740706958347951025042831, −10.22068358543928838428592717954, −9.592938614681951837097941131581, −9.563440182256676483416231456398, −8.524464693192186877471115792571, −7.19877950386906355931543697614, −7.06035091297824136120629226442, −6.16330575759644388198874314227, −5.73022578469574864023764171080, −5.04034024022624568110967431733, −4.09778591592978932596819751838, −3.10705146482163524261049807370, −2.64535399108189045264510170835, 2.64535399108189045264510170835, 3.10705146482163524261049807370, 4.09778591592978932596819751838, 5.04034024022624568110967431733, 5.73022578469574864023764171080, 6.16330575759644388198874314227, 7.06035091297824136120629226442, 7.19877950386906355931543697614, 8.524464693192186877471115792571, 9.563440182256676483416231456398, 9.592938614681951837097941131581, 10.22068358543928838428592717954, 10.66720740706958347951025042831, 12.06861790428375561600779554210, 12.18801456944399837843399013207, 12.49276997095376938063047430271, 13.18539645043385362545486359499, 13.56263174322409099297897186637, 14.13143550792508788546022118053, 14.15112524404569176674908270993

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