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Properties

Label 4-72e2-1.1-c1e2-0-3
Degree $4$
Conductor $5184$
Sign $1$
Analytic cond. $0.330536$
Root an. cond. $0.758236$
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  + 5-s + 3·7-s − 3·9-s − 5·11-s + 5·13-s − 4·17-s − 8·19-s + 23-s + 5·25-s + 9·29-s + 31-s + 3·35-s − 12·37-s − 3·41-s − 43-s − 3·45-s + 3·47-s + 7·49-s + 4·53-s − 5·55-s − 11·59-s − 7·61-s − 9·63-s + 5·65-s + 67-s + 8·71-s − 4·73-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.13·7-s − 9-s − 1.50·11-s + 1.38·13-s − 0.970·17-s − 1.83·19-s + 0.208·23-s + 25-s + 1.67·29-s + 0.179·31-s + 0.507·35-s − 1.97·37-s − 0.468·41-s − 0.152·43-s − 0.447·45-s + 0.437·47-s + 49-s + 0.549·53-s − 0.674·55-s − 1.43·59-s − 0.896·61-s − 1.13·63-s + 0.620·65-s + 0.122·67-s + 0.949·71-s − 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(0.330536\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5184,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8901687013\)
\(L(\frac12)\) \(\approx\) \(0.8901687013\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_ae
7$C_2^2$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.7.ad_c
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_o
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.af_m
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_aw
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ca
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) 2.31.ab_abe
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_abg
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_abq
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.47.ad_abm
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$C_2^2$ \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.59.l_ck
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_am
67$C_2^2$ \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \) 2.67.ab_aco
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.71.ai_gc
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) 2.79.b_ada
83$C_2^2$ \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \) 2.83.b_ade
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.89.bk_ti
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.97.an_cu
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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

−15.04596314949755698763470883040, −14.19464943602577262242666871189, −13.77072857964450116476589777167, −13.58160528459214923618533362032, −12.60509325931048725117885152786, −12.44378222695289156138008676434, −11.25386671404890912983066860805, −11.15093858439666734076360950424, −10.49873156591307052267971904906, −10.22739584430604561091129408885, −8.793596843996729558892910973048, −8.534045389151660698350575117265, −8.420518511795351127598420248194, −7.32039019467316323887690579853, −6.46187784115568010412385152438, −5.88309429526529991076949664900, −5.03124924463530726898553060118, −4.48286581092201091310744652728, −3.07728264120660582400431114899, −2.06254548776481924300575113000, 2.06254548776481924300575113000, 3.07728264120660582400431114899, 4.48286581092201091310744652728, 5.03124924463530726898553060118, 5.88309429526529991076949664900, 6.46187784115568010412385152438, 7.32039019467316323887690579853, 8.420518511795351127598420248194, 8.534045389151660698350575117265, 8.793596843996729558892910973048, 10.22739584430604561091129408885, 10.49873156591307052267971904906, 11.15093858439666734076360950424, 11.25386671404890912983066860805, 12.44378222695289156138008676434, 12.60509325931048725117885152786, 13.58160528459214923618533362032, 13.77072857964450116476589777167, 14.19464943602577262242666871189, 15.04596314949755698763470883040

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