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Properties

Label 4-1440e2-1.1-c1e2-0-41
Degree $4$
Conductor $2073600$
Sign $1$
Analytic cond. $132.214$
Root an. cond. $3.39093$
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  + 4·5-s + 4·7-s − 2·13-s + 10·17-s + 8·19-s − 4·23-s + 11·25-s + 16·35-s + 2·37-s − 12·43-s + 4·47-s + 8·49-s + 14·53-s − 8·59-s − 8·61-s − 8·65-s − 20·67-s − 6·73-s + 32·79-s + 4·83-s + 40·85-s − 8·91-s + 32·95-s − 6·97-s + 12·101-s − 12·103-s − 12·107-s + ⋯
L(s)  = 1  + 1.78·5-s + 1.51·7-s − 0.554·13-s + 2.42·17-s + 1.83·19-s − 0.834·23-s + 11/5·25-s + 2.70·35-s + 0.328·37-s − 1.82·43-s + 0.583·47-s + 8/7·49-s + 1.92·53-s − 1.04·59-s − 1.02·61-s − 0.992·65-s − 2.44·67-s − 0.702·73-s + 3.60·79-s + 0.439·83-s + 4.33·85-s − 0.838·91-s + 3.28·95-s − 0.609·97-s + 1.19·101-s − 1.18·103-s − 1.16·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2073600\)    =    \(2^{10} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(132.214\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2073600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.099811914\)
\(L(\frac12)\) \(\approx\) \(5.099811914\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_i
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.c_c
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ak_by
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_i
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_c
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_cu
47$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_i
53$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.53.ao_du
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.61.i_fi
67$C_2^2$ \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.67.u_hs
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_s
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_i
89$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.89.a_agw
97$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_s
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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

−9.639725237208135824646120450139, −9.568859221961153784326558872765, −8.956764929303487065657844439251, −8.614283874811762529455210488229, −7.906595688678972229622209256425, −7.86779084098243298372782831738, −7.31470080386650773782382785280, −7.09651170592945637339036943800, −6.15586633385312195368740810358, −6.04484554736855802635706115914, −5.44184412217736538359608171911, −5.31720677068749539916046171645, −4.86211797511930658999253024370, −4.47385662067453532093154674080, −3.44854237484596419268831412485, −3.28296345776681792012266116063, −2.49485738791724596378821822991, −2.00354792028057149372389013836, −1.32883875504121738721873184738, −1.10234636347963754766446455369, 1.10234636347963754766446455369, 1.32883875504121738721873184738, 2.00354792028057149372389013836, 2.49485738791724596378821822991, 3.28296345776681792012266116063, 3.44854237484596419268831412485, 4.47385662067453532093154674080, 4.86211797511930658999253024370, 5.31720677068749539916046171645, 5.44184412217736538359608171911, 6.04484554736855802635706115914, 6.15586633385312195368740810358, 7.09651170592945637339036943800, 7.31470080386650773782382785280, 7.86779084098243298372782831738, 7.906595688678972229622209256425, 8.614283874811762529455210488229, 8.956764929303487065657844439251, 9.568859221961153784326558872765, 9.639725237208135824646120450139

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