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Properties

Label 4-2700e2-1.1-c1e2-0-5
Degree $4$
Conductor $7290000$
Sign $1$
Analytic cond. $464.816$
Root an. cond. $4.64323$
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  − 6·11-s − 4·19-s + 12·29-s − 8·31-s + 10·49-s + 30·59-s + 10·61-s − 18·71-s + 20·79-s + 36·89-s + 12·101-s − 28·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 1.80·11-s − 0.917·19-s + 2.22·29-s − 1.43·31-s + 10/7·49-s + 3.90·59-s + 1.28·61-s − 2.13·71-s + 2.25·79-s + 3.81·89-s + 1.19·101-s − 2.68·109-s + 5/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7290000\)    =    \(2^{4} \cdot 3^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(464.816\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7290000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.654387375\)
\(L(\frac12)\) \(\approx\) \(1.654387375\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.13.a_az
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.23.a_abl
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.37.a_az
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.43.a_ade
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.47.a_adh
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.59.abe_nf
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.61.ak_fr
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.71.s_ip
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.73.a_afm
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \) 2.89.abk_ti
97$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \) 2.97.a_dr
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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

−8.822622815885386483160463810661, −8.782966421910711057101759567495, −8.156491400215480350837640150385, −7.993209601865667280851698380298, −7.56478726927075314078687412758, −7.17673414384545495847892374418, −6.69143529560710159911595894333, −6.45884779420085045227165627671, −5.92346837575260408223213354801, −5.43418354356661987532582709238, −5.06090169381853086936692023161, −4.99065431404580237332370559230, −4.17777278556012347739314747399, −3.91353820100253102888456283883, −3.38653875559862510456689753534, −2.74209343995965196925602464284, −2.29709861794426530013731719778, −2.17827964815868954922798359163, −1.08329722676254049600176654511, −0.46469455612131172375012673884, 0.46469455612131172375012673884, 1.08329722676254049600176654511, 2.17827964815868954922798359163, 2.29709861794426530013731719778, 2.74209343995965196925602464284, 3.38653875559862510456689753534, 3.91353820100253102888456283883, 4.17777278556012347739314747399, 4.99065431404580237332370559230, 5.06090169381853086936692023161, 5.43418354356661987532582709238, 5.92346837575260408223213354801, 6.45884779420085045227165627671, 6.69143529560710159911595894333, 7.17673414384545495847892374418, 7.56478726927075314078687412758, 7.993209601865667280851698380298, 8.156491400215480350837640150385, 8.782966421910711057101759567495, 8.822622815885386483160463810661

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