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Properties

Label 4-1280e2-1.1-c1e2-0-64
Degree $4$
Conductor $1638400$
Sign $1$
Analytic cond. $104.465$
Root an. cond. $3.19700$
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  − 4·3-s + 2·5-s + 4·7-s + 8·9-s − 6·13-s − 8·15-s − 6·17-s − 16·21-s − 12·23-s − 25-s − 12·27-s + 8·35-s − 6·37-s + 24·39-s − 12·43-s + 16·45-s − 12·47-s + 8·49-s + 24·51-s − 6·53-s − 16·59-s − 12·61-s + 32·63-s − 12·65-s + 12·67-s + 48·69-s + 10·73-s + ⋯
L(s)  = 1  − 2.30·3-s + 0.894·5-s + 1.51·7-s + 8/3·9-s − 1.66·13-s − 2.06·15-s − 1.45·17-s − 3.49·21-s − 2.50·23-s − 1/5·25-s − 2.30·27-s + 1.35·35-s − 0.986·37-s + 3.84·39-s − 1.82·43-s + 2.38·45-s − 1.75·47-s + 8/7·49-s + 3.36·51-s − 0.824·53-s − 2.08·59-s − 1.53·61-s + 4.03·63-s − 1.48·65-s + 1.46·67-s + 5.77·69-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1638400\)    =    \(2^{16} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(104.465\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1638400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.3.e_i
7$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_i
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_s
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.g_s
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.23.m_cu
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.29.a_acc
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_s
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_cu
47$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_cu
53$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_s
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.59.q_ha
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.67.am_cu
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ak_by
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cu
89$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.89.a_agw
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

−9.510068305118607769667969282961, −9.408887756577383664907424207566, −8.581502404159138738954688101602, −8.060088137276290538705919340885, −7.908996560457538008532018614127, −7.30227835306332592970253743319, −6.78014348839458342586090814749, −6.40657561447329141454833232388, −6.14587873090926796891878724028, −5.61326405579647884905062035838, −5.24383173031467372298135042947, −4.96278191408211701351926304571, −4.49158944607474995317860323441, −4.30475816928882914416401626860, −3.33950870353528692256487602720, −2.33522688916145357185460140694, −1.74207533632421601016936417025, −1.68375498005259898135911888355, 0, 0, 1.68375498005259898135911888355, 1.74207533632421601016936417025, 2.33522688916145357185460140694, 3.33950870353528692256487602720, 4.30475816928882914416401626860, 4.49158944607474995317860323441, 4.96278191408211701351926304571, 5.24383173031467372298135042947, 5.61326405579647884905062035838, 6.14587873090926796891878724028, 6.40657561447329141454833232388, 6.78014348839458342586090814749, 7.30227835306332592970253743319, 7.908996560457538008532018614127, 8.060088137276290538705919340885, 8.581502404159138738954688101602, 9.408887756577383664907424207566, 9.510068305118607769667969282961

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