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Properties

Label 4-4080e2-1.1-c1e2-0-7
Degree $4$
Conductor $16646400$
Sign $1$
Analytic cond. $1061.38$
Root an. cond. $5.70779$
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  − 9-s + 12·13-s − 2·17-s − 10·19-s − 25-s − 6·47-s + 13·49-s − 6·53-s − 24·59-s + 4·67-s + 81-s − 24·89-s − 32·101-s + 36·103-s − 12·117-s + 13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s + 82·169-s + ⋯
L(s)  = 1  − 1/3·9-s + 3.32·13-s − 0.485·17-s − 2.29·19-s − 1/5·25-s − 0.875·47-s + 13/7·49-s − 0.824·53-s − 3.12·59-s + 0.488·67-s + 1/9·81-s − 2.54·89-s − 3.18·101-s + 3.54·103-s − 1.10·117-s + 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.161·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16646400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1061.38\)
Root analytic conductor: \(5.70779\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16646400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.980129019\)
\(L(\frac12)\) \(\approx\) \(1.980129019\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + T^{2} \)
17$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.7.a_an
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.11.a_an
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.19.k_cl
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.29.a_abh
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.37.a_az
41$C_2^2$ \( 1 - 57 T^{2} + p^{2} T^{4} \) 2.41.a_acf
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.47.g_dz
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.53.g_el
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.61.a_aec
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.73.a_acn
79$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \) 2.79.a_afy
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.89.y_mk
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−8.495401736577312739130012422936, −8.380767276131113714660101525965, −7.992558533575272263002880497674, −7.73251612524129919563298463804, −6.93840488156627859989326230934, −6.66602605340472457982851153668, −6.44921746983497766803552943333, −5.97706923313889975833032015631, −5.84175345732411270505711862244, −5.47235090874892452706398000371, −4.68592195835422459466285296729, −4.37413804184098231457809071188, −4.02948061864655837324186087150, −3.74830044707868322484052819593, −3.11421240459696115888897648820, −2.90075701813650032475264315444, −2.04538372777742205989457203826, −1.71640747359920757115454457647, −1.22873010512347447051126994740, −0.41692476562140199595191567846, 0.41692476562140199595191567846, 1.22873010512347447051126994740, 1.71640747359920757115454457647, 2.04538372777742205989457203826, 2.90075701813650032475264315444, 3.11421240459696115888897648820, 3.74830044707868322484052819593, 4.02948061864655837324186087150, 4.37413804184098231457809071188, 4.68592195835422459466285296729, 5.47235090874892452706398000371, 5.84175345732411270505711862244, 5.97706923313889975833032015631, 6.44921746983497766803552943333, 6.66602605340472457982851153668, 6.93840488156627859989326230934, 7.73251612524129919563298463804, 7.992558533575272263002880497674, 8.380767276131113714660101525965, 8.495401736577312739130012422936

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