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Properties

Label 4-2040e2-1.1-c1e2-0-3
Degree $4$
Conductor $4161600$
Sign $1$
Analytic cond. $265.347$
Root an. cond. $4.03602$
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  − 9-s − 4·13-s + 2·17-s + 2·19-s − 25-s − 24·43-s + 2·47-s + 13·49-s + 6·53-s − 16·59-s − 4·67-s + 81-s + 16·83-s + 32·89-s − 8·101-s − 4·103-s + 4·117-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + 163-s + ⋯
L(s)  = 1  − 1/3·9-s − 1.10·13-s + 0.485·17-s + 0.458·19-s − 1/5·25-s − 3.65·43-s + 0.291·47-s + 13/7·49-s + 0.824·53-s − 2.08·59-s − 0.488·67-s + 1/9·81-s + 1.75·83-s + 3.39·89-s − 0.796·101-s − 0.394·103-s + 0.369·117-s − 0.272·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4161600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4161600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4161600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(265.347\)
Root analytic conductor: \(4.03602\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4161600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.424921772\)
\(L(\frac12)\) \(\approx\) \(1.424921772\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.11.a_d
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.29.a_aj
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \) 2.37.a_abx
41$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.41.a_ab
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.43.y_iw
47$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.47.ac_dr
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.53.ag_el
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.59.q_ha
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.67.e_fi
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.73.a_az
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.83.aq_iw
89$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.89.abg_qs
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−9.436824266306539077115739299471, −8.946849259412356435270982116753, −8.411663382880398183010252755110, −8.340763559740114388860375945700, −7.64567750442555435173031308339, −7.38601109958429232862615806097, −7.18878085217130863602039753167, −6.39714027920381571725188022968, −6.35081967552229530508156187663, −5.72866231497059758665447994214, −5.18771117995872085159798708051, −5.02357387737898518869529491446, −4.57850760931319014195924351421, −3.96285336808496329123534989509, −3.30874356180755207220779417608, −3.23671207234235473722734977833, −2.41952786674362442059026864474, −2.03142965843514793218355429492, −1.31759160324951712304812672838, −0.43571308450040052551313773415, 0.43571308450040052551313773415, 1.31759160324951712304812672838, 2.03142965843514793218355429492, 2.41952786674362442059026864474, 3.23671207234235473722734977833, 3.30874356180755207220779417608, 3.96285336808496329123534989509, 4.57850760931319014195924351421, 5.02357387737898518869529491446, 5.18771117995872085159798708051, 5.72866231497059758665447994214, 6.35081967552229530508156187663, 6.39714027920381571725188022968, 7.18878085217130863602039753167, 7.38601109958429232862615806097, 7.64567750442555435173031308339, 8.340763559740114388860375945700, 8.411663382880398183010252755110, 8.946849259412356435270982116753, 9.436824266306539077115739299471

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