Properties

Label 4-363e2-1.1-c2e2-0-0
Degree $4$
Conductor $131769$
Sign $1$
Analytic cond. $97.8325$
Root an. cond. $3.14500$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 3·4-s + 16·7-s + 27·9-s − 18·12-s − 8·13-s − 7·16-s + 12·19-s + 96·21-s + 6·25-s + 108·27-s − 48·28-s − 52·31-s − 81·36-s + 60·37-s − 48·39-s − 84·43-s − 42·48-s + 94·49-s + 24·52-s + 72·57-s − 24·61-s + 432·63-s + 69·64-s + 4·67-s + 148·73-s + 36·75-s + ⋯
L(s)  = 1  + 2·3-s − 3/4·4-s + 16/7·7-s + 3·9-s − 3/2·12-s − 0.615·13-s − 0.437·16-s + 0.631·19-s + 32/7·21-s + 6/25·25-s + 4·27-s − 1.71·28-s − 1.67·31-s − 9/4·36-s + 1.62·37-s − 1.23·39-s − 1.95·43-s − 7/8·48-s + 1.91·49-s + 6/13·52-s + 1.26·57-s − 0.393·61-s + 48/7·63-s + 1.07·64-s + 4/67·67-s + 2.02·73-s + 0.479·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(131769\)    =    \(3^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(97.8325\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{363} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 131769,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.678511298\)
\(L(\frac12)\) \(\approx\) \(5.678511298\)
\(L(2)\) 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)$
bad3$C_1$ \( ( 1 - p T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 + 3 T^{2} + p^{4} T^{4} \)
5$C_2^2$ \( 1 - 6 T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 402 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 1014 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 98 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 26 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 30 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3186 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 42 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 3018 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 2054 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 2562 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 12 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 6518 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 74 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 12194 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 1586 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 - 62 T + p^{2} T^{2} )^{2} \)
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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

−11.57004167915732868983528299016, −10.85938176637635855243928737379, −10.51396447632304208356989650948, −9.726769499814129162082651552549, −9.525196202594489687419529564589, −9.102348159074849988871042884137, −8.421573525065110104475712916289, −8.390712779255600836541277311472, −7.83190707614531253158941923857, −7.42646468162204052913179520957, −7.10510599803147858989487005421, −6.22539578064342916048976025464, −5.05633495519541799269384137125, −4.92557891552064406219353494769, −4.52576966695752003617016637773, −3.78399226838600941140469186207, −3.28550989201391704616514474527, −2.22922139084358807648094135907, −1.95828073273500628633196463102, −1.08156298363179726751424690256, 1.08156298363179726751424690256, 1.95828073273500628633196463102, 2.22922139084358807648094135907, 3.28550989201391704616514474527, 3.78399226838600941140469186207, 4.52576966695752003617016637773, 4.92557891552064406219353494769, 5.05633495519541799269384137125, 6.22539578064342916048976025464, 7.10510599803147858989487005421, 7.42646468162204052913179520957, 7.83190707614531253158941923857, 8.390712779255600836541277311472, 8.421573525065110104475712916289, 9.102348159074849988871042884137, 9.525196202594489687419529564589, 9.726769499814129162082651552549, 10.51396447632304208356989650948, 10.85938176637635855243928737379, 11.57004167915732868983528299016

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