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Properties

Label 4-72e3-1.1-c1e2-0-4
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  − 3-s + 9-s − 7·11-s + 7·17-s + 13·19-s + 6·25-s − 27-s + 7·33-s − 6·41-s + 17·43-s − 5·49-s − 7·51-s − 13·57-s − 13·59-s − 11·67-s − 7·73-s − 6·75-s + 81-s − 11·83-s + 12·89-s + 16·97-s − 7·99-s − 21·107-s + 18·113-s + 17·121-s + 6·123-s + 127-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 2.11·11-s + 1.69·17-s + 2.98·19-s + 6/5·25-s − 0.192·27-s + 1.21·33-s − 0.937·41-s + 2.59·43-s − 5/7·49-s − 0.980·51-s − 1.72·57-s − 1.69·59-s − 1.34·67-s − 0.819·73-s − 0.692·75-s + 1/9·81-s − 1.20·83-s + 1.27·89-s + 1.62·97-s − 0.703·99-s − 2.03·107-s + 1.69·113-s + 1.54·121-s + 0.541·123-s + 0.0887·127-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

\(L(1)\) \(\approx\) \(1.446248043\)
\(L(\frac12)\) \(\approx\) \(1.446248043\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.11.h_bg
13$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.13.a_e
17$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ah_bs
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.19.an_da
23$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \) 2.23.a_j
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.29.a_aby
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.a_bl
37$C_2^2$ \( 1 - 52 T^{2} + p^{2} T^{4} \) 2.37.a_aca
41$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.g_cd
43$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.43.ar_ga
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.53.a_aco
59$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.n_gc
61$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.61.a_ao
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.67.l_ee
71$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.71.a_ada
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.73.h_ga
79$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.79.a_ack
83$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.83.l_fk
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.am_gw
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.aq_io
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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

−8.689469098016838617603663115960, −7.980967822814334950460501615076, −7.70028465500369463432694640124, −7.36667224515467056263681932491, −7.12756647082018857777826798579, −6.09250801146793651926355859566, −5.79229517863849518880882057975, −5.33838938220385141983366846937, −5.00828683194109319487367368965, −4.57325676586049821824746001737, −3.53788677262904603844826991051, −3.00648522025045462679203043425, −2.78233382743057856226979304936, −1.50680724926070052066489735521, −0.74842540777724196576532462686, 0.74842540777724196576532462686, 1.50680724926070052066489735521, 2.78233382743057856226979304936, 3.00648522025045462679203043425, 3.53788677262904603844826991051, 4.57325676586049821824746001737, 5.00828683194109319487367368965, 5.33838938220385141983366846937, 5.79229517863849518880882057975, 6.09250801146793651926355859566, 7.12756647082018857777826798579, 7.36667224515467056263681932491, 7.70028465500369463432694640124, 7.980967822814334950460501615076, 8.689469098016838617603663115960

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