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Properties

Label 4-733e2-1.1-c1e2-0-0
Degree $4$
Conductor $537289$
Sign $1$
Analytic cond. $34.2580$
Root an. cond. $2.41930$
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-s + 4-s + 3·7-s + 3·9-s + 12-s − 9·13-s − 3·16-s − 3·17-s + 8·19-s + 3·21-s + 15·23-s + 10·25-s + 8·27-s + 3·28-s − 5·31-s + 3·36-s − 11·37-s − 9·39-s − 3·41-s + 9·43-s − 3·48-s − 49-s − 3·51-s − 9·52-s − 3·53-s + 8·57-s + 3·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 1.13·7-s + 9-s + 0.288·12-s − 2.49·13-s − 3/4·16-s − 0.727·17-s + 1.83·19-s + 0.654·21-s + 3.12·23-s + 2·25-s + 1.53·27-s + 0.566·28-s − 0.898·31-s + 1/2·36-s − 1.80·37-s − 1.44·39-s − 0.468·41-s + 1.37·43-s − 0.433·48-s − 1/7·49-s − 0.420·51-s − 1.24·52-s − 0.412·53-s + 1.05·57-s + 0.390·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 537289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 537289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(537289\)    =    \(733^{2}\)
Sign: $1$
Analytic conductor: \(34.2580\)
Root analytic conductor: \(2.41930\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 537289,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.338199184\)
\(L(\frac12)\) \(\approx\) \(3.338199184\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad733$C_2$ \( 1 + 50 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_ac
5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.ad_k
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.j_bo
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 - 15 T + 98 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.23.ap_du
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_ag
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.l_dg
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_abg
43$C_2^2$ \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.43.aj_cs
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_abs
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.59.ad_aby
61$C_2^2$ \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.61.l_ci
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.a_aes
71$C_2^2$ \( 1 + 9 T + 98 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.71.j_du
73$C_2^2$ \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.73.an_ds
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ada
83$C_2^2$ \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.83.ad_acw
89$C_2^2$ \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.89.aj_ai
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.af_acu
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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

−10.78377970273907219253692813652, −10.29509219875896800510453785081, −9.463415551565282849700038220838, −9.397178829606554178172186058524, −8.912969942857574020244819670513, −8.627809487322859616127179145827, −7.87124804318195334523724994456, −7.28424320060379153650710468875, −7.21455699599743039905521877785, −7.01183242683288923836552704039, −6.49974328300329451971903432564, −5.29237342508406872728026759820, −5.05574100068455474183293127021, −4.79874725293264180007510510237, −4.48695921995988133997616854413, −3.29102066287059034490869218491, −2.96978514113318081338875493507, −2.44454355038518330941036293003, −1.72695758687764395728473617590, −0.982202205968868494059130611212, 0.982202205968868494059130611212, 1.72695758687764395728473617590, 2.44454355038518330941036293003, 2.96978514113318081338875493507, 3.29102066287059034490869218491, 4.48695921995988133997616854413, 4.79874725293264180007510510237, 5.05574100068455474183293127021, 5.29237342508406872728026759820, 6.49974328300329451971903432564, 7.01183242683288923836552704039, 7.21455699599743039905521877785, 7.28424320060379153650710468875, 7.87124804318195334523724994456, 8.627809487322859616127179145827, 8.912969942857574020244819670513, 9.397178829606554178172186058524, 9.463415551565282849700038220838, 10.29509219875896800510453785081, 10.78377970273907219253692813652

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