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Properties

Label 4-20e4-1.1-c1e2-0-21
Degree $4$
Conductor $160000$
Sign $1$
Analytic cond. $10.2017$
Root an. cond. $1.78718$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·3-s + 2·4-s + 8·6-s + 6·7-s + 6·9-s − 2·11-s + 8·12-s + 12·14-s − 4·16-s − 2·17-s + 12·18-s − 6·19-s + 24·21-s − 4·22-s + 2·23-s − 4·27-s + 12·28-s − 14·29-s − 8·32-s − 8·33-s − 4·34-s + 12·36-s − 12·38-s + 48·42-s − 4·44-s + 4·46-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 4-s + 3.26·6-s + 2.26·7-s + 2·9-s − 0.603·11-s + 2.30·12-s + 3.20·14-s − 16-s − 0.485·17-s + 2.82·18-s − 1.37·19-s + 5.23·21-s − 0.852·22-s + 0.417·23-s − 0.769·27-s + 2.26·28-s − 2.59·29-s − 1.41·32-s − 1.39·33-s − 0.685·34-s + 2·36-s − 1.94·38-s + 7.40·42-s − 0.603·44-s + 0.589·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(160000\)    =    \(2^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(10.2017\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.782084086\)
\(L(\frac12)\) \(\approx\) \(7.782084086\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.3.ae_k
7$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.7.ag_s
11$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_c
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_c
19$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_s
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_c
29$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.o_du
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.31.a_acg
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.37.a_abm
41$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.41.a_aco
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.47.ao_du
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.53.aq_go
59$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_s
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.c_c
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
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.aq_io
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.83.ae_go
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2^2$ \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.97.aw_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

−11.35455486396953437560263081247, −11.34271634361405545287847666739, −10.73379146969660522359578918770, −10.27849030039582616896656543486, −9.295031495544919730835383000604, −9.080446070871572926049573007329, −8.539389580796878065360657991126, −8.500356497166096828363217037631, −7.70025707207685836442379056303, −7.56388854713531113647021984607, −7.03828459964845757621321930462, −6.02602324364513592661050955030, −5.50673183009769169056697951338, −5.12888088373373741374021003178, −4.37675151236705471912632784709, −3.98369903853993104647122963835, −3.57945930056259653910673692393, −2.66300934111373105017416825501, −2.15551292005125734328346292541, −1.97465207062418331185844996016, 1.97465207062418331185844996016, 2.15551292005125734328346292541, 2.66300934111373105017416825501, 3.57945930056259653910673692393, 3.98369903853993104647122963835, 4.37675151236705471912632784709, 5.12888088373373741374021003178, 5.50673183009769169056697951338, 6.02602324364513592661050955030, 7.03828459964845757621321930462, 7.56388854713531113647021984607, 7.70025707207685836442379056303, 8.500356497166096828363217037631, 8.539389580796878065360657991126, 9.080446070871572926049573007329, 9.295031495544919730835383000604, 10.27849030039582616896656543486, 10.73379146969660522359578918770, 11.34271634361405545287847666739, 11.35455486396953437560263081247

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