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Properties

Label 4-72e3-1.1-c1e2-0-10
Degree $4$
Conductor $373248$
Sign $1$
Analytic cond. $23.7986$
Root an. cond. $2.20870$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·11-s + 2·13-s + 4·23-s − 2·25-s + 2·37-s − 4·47-s − 49-s + 12·59-s + 2·61-s + 16·71-s + 6·73-s − 8·83-s + 6·97-s − 4·107-s − 12·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 8·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 19·169-s + ⋯
L(s)  = 1  + 1.20·11-s + 0.554·13-s + 0.834·23-s − 2/5·25-s + 0.328·37-s − 0.583·47-s − 1/7·49-s + 1.56·59-s + 0.256·61-s + 1.89·71-s + 0.702·73-s − 0.878·83-s + 0.609·97-s − 0.386·107-s − 1.14·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.668·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.46·169-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: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 373248,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.086584579\)
\(L(\frac12)\) \(\approx\) \(2.086584579\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.7.a_b
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.ac_x
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.19.a_f
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.ae_bi
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.31.a_abi
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.37.ac_ct
41$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.41.a_be
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.e_de
53$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.53.a_cc
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.61.ac_dj
67$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.67.a_ad
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.aq_fm
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.73.ag_dn
79$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \) 2.79.a_cn
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.i_eo
89$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.89.a_cg
97$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.97.ag_hf
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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

−8.743912193418616822864910425073, −8.184320786590837395942904402833, −7.941676071030135990913003078896, −7.16223801271259372079253947760, −6.81469679352057428554685304429, −6.44853603535450955720124481511, −5.92423783741422435038698284803, −5.33831608143869498911430837835, −4.90152044564084412247796703588, −4.12420574187993066594736002836, −3.82654759034938098041962994491, −3.21886801647729298479210530147, −2.45674529157672829054902809510, −1.65457487863733924887715004440, −0.869770728637159829995538902484, 0.869770728637159829995538902484, 1.65457487863733924887715004440, 2.45674529157672829054902809510, 3.21886801647729298479210530147, 3.82654759034938098041962994491, 4.12420574187993066594736002836, 4.90152044564084412247796703588, 5.33831608143869498911430837835, 5.92423783741422435038698284803, 6.44853603535450955720124481511, 6.81469679352057428554685304429, 7.16223801271259372079253947760, 7.941676071030135990913003078896, 8.184320786590837395942904402833, 8.743912193418616822864910425073

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