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Properties

Label 4-3096e2-1.1-c1e2-0-3
Degree $4$
Conductor $9585216$
Sign $1$
Analytic cond. $611.161$
Root an. cond. $4.97209$
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  − 4·5-s + 2·25-s + 12·29-s − 4·31-s − 10·43-s + 14·49-s + 28·67-s + 24·71-s + 20·79-s − 28·89-s − 4·97-s + 12·109-s + 20·113-s + 20·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + 149-s + 151-s + 16·155-s + 157-s + 163-s + 167-s − 26·169-s + ⋯
L(s)  = 1  − 1.78·5-s + 2/5·25-s + 2.22·29-s − 0.718·31-s − 1.52·43-s + 2·49-s + 3.42·67-s + 2.84·71-s + 2.25·79-s − 2.96·89-s − 0.406·97-s + 1.14·109-s + 1.88·113-s + 1.81·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + 0.0819·149-s + 0.0813·151-s + 1.28·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9585216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9585216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9585216\)    =    \(2^{6} \cdot 3^{4} \cdot 43^{2}\)
Sign: $1$
Analytic conductor: \(611.161\)
Root analytic conductor: \(4.97209\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9585216,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.768102895\)
\(L(\frac12)\) \(\approx\) \(1.768102895\)
\(L(\frac{3}{2})\) 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 \( 1 \)
43$C_2$ \( 1 + 10 T + p T^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.11.a_au
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.17.a_q
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.23.a_abs
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.37.a_ac
41$C_2^2$ \( 1 - 64 T^{2} + p^{2} T^{4} \) 2.41.a_acm
47$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \) 2.47.a_ado
53$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \) 2.53.a_aea
59$C_2^2$ \( 1 - 100 T^{2} + p^{2} T^{4} \) 2.59.a_adw
61$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.61.a_aek
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.67.abc_ms
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.73.a_acw
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.83.a_ae
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.89.bc_ok
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−8.504086131249739847584053917668, −8.486845686518032699732768737330, −8.078771650454477985715992289794, −8.045725228202635196573269030095, −7.21929179656618413371794228214, −7.18882980138774063014594526600, −6.71384251073176095385230425537, −6.45125772872042624964667559259, −5.66582579397380973429946905057, −5.56430929533400338359064093723, −4.82887417262458704273129734312, −4.67138778513838048227146681925, −4.00689350101133140454196853445, −3.92525942810805146726065214759, −3.20856385959482307972463412995, −3.19822534735738129371931262492, −2.16179179194181853578539257065, −2.02435442355053038638777075840, −0.74553206109740259248772397539, −0.64096002281387407379303769302, 0.64096002281387407379303769302, 0.74553206109740259248772397539, 2.02435442355053038638777075840, 2.16179179194181853578539257065, 3.19822534735738129371931262492, 3.20856385959482307972463412995, 3.92525942810805146726065214759, 4.00689350101133140454196853445, 4.67138778513838048227146681925, 4.82887417262458704273129734312, 5.56430929533400338359064093723, 5.66582579397380973429946905057, 6.45125772872042624964667559259, 6.71384251073176095385230425537, 7.18882980138774063014594526600, 7.21929179656618413371794228214, 8.045725228202635196573269030095, 8.078771650454477985715992289794, 8.486845686518032699732768737330, 8.504086131249739847584053917668

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