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Properties

Label 4-7440e2-1.1-c1e2-0-9
Degree $4$
Conductor $55353600$
Sign $1$
Analytic cond. $3529.39$
Root an. cond. $7.70770$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s + 4·7-s + 3·9-s − 8·13-s − 4·15-s − 8·21-s + 4·23-s + 3·25-s − 4·27-s − 12·29-s + 2·31-s + 8·35-s + 16·39-s − 4·41-s − 8·43-s + 6·45-s + 12·47-s − 16·53-s − 8·59-s − 4·61-s + 12·63-s − 16·65-s + 4·67-s − 8·69-s + 8·71-s − 16·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 1.51·7-s + 9-s − 2.21·13-s − 1.03·15-s − 1.74·21-s + 0.834·23-s + 3/5·25-s − 0.769·27-s − 2.22·29-s + 0.359·31-s + 1.35·35-s + 2.56·39-s − 0.624·41-s − 1.21·43-s + 0.894·45-s + 1.75·47-s − 2.19·53-s − 1.04·59-s − 0.512·61-s + 1.51·63-s − 1.98·65-s + 0.488·67-s − 0.963·69-s + 0.949·71-s − 1.87·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(55353600\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(3529.39\)
Root analytic conductor: \(7.70770\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 55353600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
31$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_q
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.13.i_bo
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.17.a_ba
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.19.a_be
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_s
29$D_{4}$ \( 1 + 12 T + 76 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.29.m_cy
37$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \) 2.37.a_y
41$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_da
43$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_cs
47$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_du
53$D_{4}$ \( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.53.q_gg
59$D_{4}$ \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_fc
61$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_eo
67$D_{4}$ \( 1 - 4 T + 88 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_dk
71$D_{4}$ \( 1 - 8 T + 156 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_ga
73$D_{4}$ \( 1 + 16 T + 208 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.73.q_ia
79$D_{4}$ \( 1 + 12 T + 186 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_he
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.83.ae_go
89$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.89.e_u
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_gk
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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

−7.48192866451101318573492523098, −7.47992356493877880056743157297, −6.90435878409197978226304711528, −6.87209543893531896621859969127, −6.10517709485794166209173783998, −6.05840610200207749779675478306, −5.47317463810225794773878574088, −5.15847595157747234705151284658, −4.99030374446178257749325204399, −4.83577617162774262015549969054, −4.23609542244433283403453054261, −4.04746205016165923444760742539, −3.14476262103086583100069967605, −2.96285831919115570156810940628, −2.14349453250815963081491470228, −2.09332414043169185212758718916, −1.34570222921733850703497913296, −1.31714143326460555390295654810, 0, 0, 1.31714143326460555390295654810, 1.34570222921733850703497913296, 2.09332414043169185212758718916, 2.14349453250815963081491470228, 2.96285831919115570156810940628, 3.14476262103086583100069967605, 4.04746205016165923444760742539, 4.23609542244433283403453054261, 4.83577617162774262015549969054, 4.99030374446178257749325204399, 5.15847595157747234705151284658, 5.47317463810225794773878574088, 6.05840610200207749779675478306, 6.10517709485794166209173783998, 6.87209543893531896621859969127, 6.90435878409197978226304711528, 7.47992356493877880056743157297, 7.48192866451101318573492523098

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