↑

Properties

Label 4-4032e2-1.1-c1e2-0-10
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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s + 12·17-s − 16·23-s + 6·25-s + 16·31-s − 12·41-s − 24·47-s + 3·49-s + 12·73-s + 12·89-s − 20·97-s − 8·103-s + 36·113-s + 24·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 32·161-s + 163-s + 167-s − 10·169-s + 173-s + ⋯
L(s)  = 1  + 0.755·7-s + 2.91·17-s − 3.33·23-s + 6/5·25-s + 2.87·31-s − 1.87·41-s − 3.50·47-s + 3/7·49-s + 1.40·73-s + 1.27·89-s − 2.03·97-s − 0.788·103-s + 3.38·113-s + 2.20·119-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 − 2.52·161-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-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

\(L(1)\) \(\approx\) \(2.856040097\)
\(L(\frac12)\) \(\approx\) \(2.856040097\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.19.a_ba
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.23.q_eg
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
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$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.83.a_ady
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.523314379397523693242547652349, −8.107806387445461734908194411798, −7.969095571795307483864421522974, −7.81432805183976559464225862439, −7.24645089593918393175211757276, −6.60557678119049794964677576086, −6.52351628240022775683468020893, −5.91747212544699068338250343257, −5.83059751364762622630481314685, −5.09679738832542045696119401124, −4.94652862760079836649269118425, −4.62278395114556380682361410572, −4.01516792086914401998502170719, −3.44120147904950181444688071969, −3.38820289370264574518715979778, −2.74756526346496577955245762443, −2.18741415405144544660374849034, −1.54555916467249569238410133011, −1.30419564962140819913175798835, −0.50700070140376150295088311047, 0.50700070140376150295088311047, 1.30419564962140819913175798835, 1.54555916467249569238410133011, 2.18741415405144544660374849034, 2.74756526346496577955245762443, 3.38820289370264574518715979778, 3.44120147904950181444688071969, 4.01516792086914401998502170719, 4.62278395114556380682361410572, 4.94652862760079836649269118425, 5.09679738832542045696119401124, 5.83059751364762622630481314685, 5.91747212544699068338250343257, 6.52351628240022775683468020893, 6.60557678119049794964677576086, 7.24645089593918393175211757276, 7.81432805183976559464225862439, 7.969095571795307483864421522974, 8.107806387445461734908194411798, 8.523314379397523693242547652349

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