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Properties

Label 4-273e2-1.1-c1e2-0-3
Degree $4$
Conductor $74529$
Sign $1$
Analytic cond. $4.75203$
Root an. cond. $1.47645$
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  − 3·2-s + 3-s + 4·4-s − 6·5-s − 3·6-s + 7-s − 3·8-s + 18·10-s + 9·11-s + 4·12-s − 2·13-s − 3·14-s − 6·15-s + 3·16-s + 3·17-s + 9·19-s − 24·20-s + 21-s − 27·22-s − 6·23-s − 3·24-s + 19·25-s + 6·26-s − 27-s + 4·28-s + 6·29-s + 18·30-s + ⋯
L(s)  = 1  − 2.12·2-s + 0.577·3-s + 2·4-s − 2.68·5-s − 1.22·6-s + 0.377·7-s − 1.06·8-s + 5.69·10-s + 2.71·11-s + 1.15·12-s − 0.554·13-s − 0.801·14-s − 1.54·15-s + 3/4·16-s + 0.727·17-s + 2.06·19-s − 5.36·20-s + 0.218·21-s − 5.75·22-s − 1.25·23-s − 0.612·24-s + 19/5·25-s + 1.17·26-s − 0.192·27-s + 0.755·28-s + 1.11·29-s + 3.28·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(74529\)    =    \(3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(4.75203\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 74529,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3750007014\)
\(L(\frac12)\) \(\approx\) \(0.3750007014\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.2.d_f
5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.5.g_r
11$C_2^2$ \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.11.aj_bm
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.19.aj_bu
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_n
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.29.ag_cp
31$C_2^2$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_br
37$C_2^2$ \( 1 + 6 T + 49 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_bx
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.41.a_ba
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2^2$ \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.47.d_by
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_abs
59$C_2^2$ \( 1 + 15 T + 134 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.59.p_fe
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.b_aci
67$C_2^2$ \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.67.ad_cs
71$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.71.a_f
73$C_2^2$ \( 1 + 18 T + 181 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.73.s_gz
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.79.e_acl
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \) 2.83.a_afy
89$C_2^2$ \( 1 - 6 T + 101 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_dx
97$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \) 2.97.a_ec
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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

−11.84440254993144107741731882980, −11.76371530990473497010391220833, −11.23718717750221031381516919891, −10.67513614046649055393601100204, −9.819760625622198764040560566659, −9.761749774580147640585612782666, −9.121872411717637675563979785834, −8.683858897455107663618442120565, −8.445724025336055315310301250677, −7.73894943230098773934158074681, −7.65497830786548578635213910945, −7.31011543516290121031769716270, −6.58677003388973727867381772807, −5.85173327134042697677499112049, −4.51359882383000845563905513325, −4.33954555862275919588376019151, −3.35632158410707041133788598801, −3.27596609173064951824024846724, −1.43928483439780298724367027967, −0.74559322105855234028139010698, 0.74559322105855234028139010698, 1.43928483439780298724367027967, 3.27596609173064951824024846724, 3.35632158410707041133788598801, 4.33954555862275919588376019151, 4.51359882383000845563905513325, 5.85173327134042697677499112049, 6.58677003388973727867381772807, 7.31011543516290121031769716270, 7.65497830786548578635213910945, 7.73894943230098773934158074681, 8.445724025336055315310301250677, 8.683858897455107663618442120565, 9.121872411717637675563979785834, 9.761749774580147640585612782666, 9.819760625622198764040560566659, 10.67513614046649055393601100204, 11.23718717750221031381516919891, 11.76371530990473497010391220833, 11.84440254993144107741731882980

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