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Properties

Label 4-855e2-1.1-c1e2-0-9
Degree $4$
Conductor $731025$
Sign $1$
Analytic cond. $46.6107$
Root an. cond. $2.61289$
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  + 3·4-s + 2·5-s + 8·11-s + 5·16-s − 2·19-s + 6·20-s − 25-s − 12·29-s − 8·31-s + 20·41-s + 24·44-s + 10·49-s + 16·55-s + 4·61-s + 3·64-s − 8·71-s − 6·76-s − 8·79-s + 10·80-s − 4·89-s − 4·95-s − 3·100-s + 12·101-s − 12·109-s − 36·116-s + 26·121-s − 24·124-s + ⋯
L(s)  = 1  + 3/2·4-s + 0.894·5-s + 2.41·11-s + 5/4·16-s − 0.458·19-s + 1.34·20-s − 1/5·25-s − 2.22·29-s − 1.43·31-s + 3.12·41-s + 3.61·44-s + 10/7·49-s + 2.15·55-s + 0.512·61-s + 3/8·64-s − 0.949·71-s − 0.688·76-s − 0.900·79-s + 1.11·80-s − 0.423·89-s − 0.410·95-s − 0.299·100-s + 1.19·101-s − 1.14·109-s − 3.34·116-s + 2.36·121-s − 2.15·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(731025\)    =    \(3^{4} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(46.6107\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 731025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.280194459\)
\(L(\frac12)\) \(\approx\) \(4.280194459\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.43.a_ade
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.47.a_acg
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.53.a_ag
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.71.i_gc
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \) 2.83.a_gc
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \) 2.97.a_agc
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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

−10.58708562452459998415874618689, −9.895725449970585367293151930639, −9.378141900917385247922674040582, −9.272490544274385803527697612288, −8.956817079276231557468572180551, −8.300398733983534919188685853658, −7.53486084582928610991960209872, −7.32920404736049357396229373510, −7.01032123073199462453871565352, −6.39914183394441507505221523728, −6.13934856384532493371442001671, −5.63858215501018800429036408746, −5.53515279053185991094048963791, −4.20488842804589520322511362464, −4.16123666520399111066634718381, −3.51404870583363007594206669721, −2.79360950708595798475222006793, −1.96629034009164122585877834940, −1.90450570379301442420146239718, −1.07559288457100877122833086134, 1.07559288457100877122833086134, 1.90450570379301442420146239718, 1.96629034009164122585877834940, 2.79360950708595798475222006793, 3.51404870583363007594206669721, 4.16123666520399111066634718381, 4.20488842804589520322511362464, 5.53515279053185991094048963791, 5.63858215501018800429036408746, 6.13934856384532493371442001671, 6.39914183394441507505221523728, 7.01032123073199462453871565352, 7.32920404736049357396229373510, 7.53486084582928610991960209872, 8.300398733983534919188685853658, 8.956817079276231557468572180551, 9.272490544274385803527697612288, 9.378141900917385247922674040582, 9.895725449970585367293151930639, 10.58708562452459998415874618689

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