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Properties

Label 4-4032e2-1.1-c1e2-0-14
Degree $4$
Conductor $16257024$
Sign $1$
Analytic cond. $1036.56$
Root an. cond. $5.67412$
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  − 2·7-s − 4·17-s + 6·25-s + 16·31-s + 4·41-s − 8·47-s + 3·49-s + 16·71-s + 12·73-s + 32·79-s + 28·89-s − 20·97-s + 8·103-s − 28·113-s + 8·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯
L(s)  = 1  − 0.755·7-s − 0.970·17-s + 6/5·25-s + 2.87·31-s + 0.624·41-s − 1.16·47-s + 3/7·49-s + 1.89·71-s + 1.40·73-s + 3.60·79-s + 2.96·89-s − 2.03·97-s + 0.788·103-s − 2.63·113-s + 0.733·119-s + 1.63·121-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 + 1.69·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16257024\)    =    \(2^{12} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1036.56\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16257024,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.708010613\)
\(L(\frac12)\) \(\approx\) \(2.708010613\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.37.a_acg
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.47.i_eg
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.59.a_ady
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.61.a_aeo
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.89.abc_ok
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−8.575686048537365608585292180295, −8.234478001984109873978363518907, −7.87033671697521534918003061363, −7.73556119101459360541781670367, −6.97396435540873469297528334742, −6.59106095295739766102385936769, −6.43352084217862655939081795922, −6.42490356413633172068184548476, −5.67308861390616284562900152792, −5.15458292471109153400824436071, −4.82143923734404707735838866992, −4.64052140343595092000810765717, −3.90958286248636377622440988773, −3.73445382354832769524019201901, −3.07558292377551450936890536266, −2.75313294668644967488786812973, −2.32574174193256105547932828870, −1.80592410161092673690940912608, −0.855076024767812872077051289248, −0.63122428606590345487515118807, 0.63122428606590345487515118807, 0.855076024767812872077051289248, 1.80592410161092673690940912608, 2.32574174193256105547932828870, 2.75313294668644967488786812973, 3.07558292377551450936890536266, 3.73445382354832769524019201901, 3.90958286248636377622440988773, 4.64052140343595092000810765717, 4.82143923734404707735838866992, 5.15458292471109153400824436071, 5.67308861390616284562900152792, 6.42490356413633172068184548476, 6.43352084217862655939081795922, 6.59106095295739766102385936769, 6.97396435540873469297528334742, 7.73556119101459360541781670367, 7.87033671697521534918003061363, 8.234478001984109873978363518907, 8.575686048537365608585292180295

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