Properties

Label 4-231e2-1.1-c1e2-0-2
Degree $4$
Conductor $53361$
Sign $1$
Analytic cond. $3.40234$
Root an. cond. $1.35814$
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  − 2-s + 3-s + 2·4-s − 6-s − 4·7-s − 5·8-s + 11-s + 2·12-s + 8·13-s + 4·14-s + 5·16-s − 3·17-s − 19-s − 4·21-s − 22-s + 23-s − 5·24-s + 5·25-s − 8·26-s − 27-s − 8·28-s − 10·29-s + 10·31-s − 10·32-s + 33-s + 3·34-s + 11·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 4-s − 0.408·6-s − 1.51·7-s − 1.76·8-s + 0.301·11-s + 0.577·12-s + 2.21·13-s + 1.06·14-s + 5/4·16-s − 0.727·17-s − 0.229·19-s − 0.872·21-s − 0.213·22-s + 0.208·23-s − 1.02·24-s + 25-s − 1.56·26-s − 0.192·27-s − 1.51·28-s − 1.85·29-s + 1.79·31-s − 1.76·32-s + 0.174·33-s + 0.514·34-s + 1.80·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(53361\)    =    \(3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3.40234\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 53361,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.122287960\)
\(L(\frac12)\) \(\approx\) \(1.122287960\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
11$C_2$ \( 1 - T + T^{2} \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 8 T + 11 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−12.56260010464233497128926631783, −11.61955493841035508480166779106, −11.59966217936040048271387732655, −10.90984236539440861779799700846, −10.55394055914604955629829112434, −9.963700770653904185328040249705, −9.083560998572898870980188836567, −9.080612673425617263878261001329, −8.963440169079500021874441588979, −8.011268862546300472820306804149, −7.60776879386542406951634831054, −6.58345711018804532854311212549, −6.58222026270975518218619741051, −6.03716599382708928091827731224, −5.57757295225558605256236355966, −3.96352857462802297745474766550, −3.84181726249563403193232508855, −2.71144282454676931425179080320, −2.64142776038863682213423741847, −0.992788723954987033963736789232, 0.992788723954987033963736789232, 2.64142776038863682213423741847, 2.71144282454676931425179080320, 3.84181726249563403193232508855, 3.96352857462802297745474766550, 5.57757295225558605256236355966, 6.03716599382708928091827731224, 6.58222026270975518218619741051, 6.58345711018804532854311212549, 7.60776879386542406951634831054, 8.011268862546300472820306804149, 8.963440169079500021874441588979, 9.080612673425617263878261001329, 9.083560998572898870980188836567, 9.963700770653904185328040249705, 10.55394055914604955629829112434, 10.90984236539440861779799700846, 11.59966217936040048271387732655, 11.61955493841035508480166779106, 12.56260010464233497128926631783

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