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Properties

Label 4-4805e2-1.1-c1e2-0-0
Degree $4$
Conductor $23088025$
Sign $1$
Analytic cond. $1472.11$
Root an. cond. $6.19420$
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 − 4-s − 2·5-s − 4·7-s + 8·8-s − 4·9-s + 4·10-s + 8·14-s − 7·16-s + 8·18-s − 8·19-s + 2·20-s + 3·25-s + 4·28-s − 14·32-s + 8·35-s + 4·36-s + 16·38-s − 16·40-s + 20·41-s + 8·45-s − 2·49-s − 6·50-s − 32·56-s + 16·59-s + 16·63-s + 35·64-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s − 0.894·5-s − 1.51·7-s + 2.82·8-s − 4/3·9-s + 1.26·10-s + 2.13·14-s − 7/4·16-s + 1.88·18-s − 1.83·19-s + 0.447·20-s + 3/5·25-s + 0.755·28-s − 2.47·32-s + 1.35·35-s + 2/3·36-s + 2.59·38-s − 2.52·40-s + 3.12·41-s + 1.19·45-s − 2/7·49-s − 0.848·50-s − 4.27·56-s + 2.08·59-s + 2.01·63-s + 35/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23088025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23088025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23088025\)    =    \(5^{2} \cdot 31^{4}\)
Sign: $1$
Analytic conductor: \(1472.11\)
Root analytic conductor: \(6.19420\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23088025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1040015871\)
\(L(\frac12)\) \(\approx\) \(0.1040015871\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad5$C_1$ \( ( 1 + T )^{2} \)
31 \( 1 \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.2.c_f
3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.23.a_ae
29$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.29.a_i
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.37.a_c
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
43$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \) 2.43.a_cq
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.53.a_bi
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.59.aq_ha
61$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \) 2.61.a_cu
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.71.i_gc
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.73.a_cw
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \) 2.83.a_em
89$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.89.a_dc
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−8.509702196785107422645825564399, −8.401930384415265433384194663249, −7.68073952698784522091600676042, −7.67971646231990049055635960607, −7.28162700929389523460152344321, −6.71686470700385356708667140741, −6.24724845939990654032938259012, −6.20578143624530865174296757944, −5.43905562119782544669449151032, −5.28767645266726421929097867133, −4.58945096079295981989720501552, −4.34213532244743429247085747350, −3.83850129746067797882034427780, −3.74825718470574142532206638617, −3.06938626372023333778866747615, −2.56999900596291065924702035050, −2.19384884798111953443839211330, −1.29265327702840687191771619909, −0.65501376125130863335748731147, −0.20216187348208983450575852366, 0.20216187348208983450575852366, 0.65501376125130863335748731147, 1.29265327702840687191771619909, 2.19384884798111953443839211330, 2.56999900596291065924702035050, 3.06938626372023333778866747615, 3.74825718470574142532206638617, 3.83850129746067797882034427780, 4.34213532244743429247085747350, 4.58945096079295981989720501552, 5.28767645266726421929097867133, 5.43905562119782544669449151032, 6.20578143624530865174296757944, 6.24724845939990654032938259012, 6.71686470700385356708667140741, 7.28162700929389523460152344321, 7.67971646231990049055635960607, 7.68073952698784522091600676042, 8.401930384415265433384194663249, 8.509702196785107422645825564399

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