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Properties

Label 4-72e3-1.1-c1e2-0-1
Degree $4$
Conductor $373248$
Sign $1$
Analytic cond. $23.7986$
Root an. cond. $2.20870$
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  − 2·5-s + 4·19-s + 4·23-s − 7·25-s + 12·29-s − 4·43-s + 12·47-s − 5·49-s + 10·53-s − 20·67-s − 16·71-s + 2·73-s − 8·95-s − 2·97-s + 18·101-s − 8·115-s + 3·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.917·19-s + 0.834·23-s − 7/5·25-s + 2.22·29-s − 0.609·43-s + 1.75·47-s − 5/7·49-s + 1.37·53-s − 2.44·67-s − 1.89·71-s + 0.234·73-s − 0.820·95-s − 0.203·97-s + 1.79·101-s − 0.746·115-s + 3/11·121-s + 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(373248\)    =    \(2^{9} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(23.7986\)
Root analytic conductor: \(2.20870\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 373248,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.401397445\)
\(L(\frac12)\) \(\approx\) \(1.401397445\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.5.c_l
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.11.a_ad
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.23.ae_by
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.a_n
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.43.e_dm
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.53.ak_fb
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.67.u_ja
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.73.ac_fr
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.83.a_bt
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.97.c_hn
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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

−8.554704477910262490194224105907, −8.352104784585146468182630520222, −7.67996846591970940803333630612, −7.25448367711369655898353657421, −7.15340015105356378963058851080, −6.23138020629374792100300833919, −5.98276615954030285758204092167, −5.32932694357691635729819844111, −4.71525232702415007486392238194, −4.34190910531857724149474492775, −3.74514947909231141400939990651, −3.11556422792398663255142778514, −2.66387606752889082799803334542, −1.64081961055094255152301138209, −0.68688623656136131006877383651, 0.68688623656136131006877383651, 1.64081961055094255152301138209, 2.66387606752889082799803334542, 3.11556422792398663255142778514, 3.74514947909231141400939990651, 4.34190910531857724149474492775, 4.71525232702415007486392238194, 5.32932694357691635729819844111, 5.98276615954030285758204092167, 6.23138020629374792100300833919, 7.15340015105356378963058851080, 7.25448367711369655898353657421, 7.67996846591970940803333630612, 8.352104784585146468182630520222, 8.554704477910262490194224105907

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