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Properties

Label 4-2e2-1.1-c47e2-0-0
Degree $4$
Conductor $4$
Sign $1$
Analytic cond. $782.966$
Root an. cond. $5.28975$
Motivic weight $47$
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  − 1.67e7·2-s + 1.22e11·3-s + 2.11e14·4-s + 1.81e16·5-s − 2.05e18·6-s + 1.61e20·7-s − 2.36e21·8-s + 2.06e21·9-s − 3.04e23·10-s + 1.14e24·11-s + 2.58e25·12-s − 9.11e25·13-s − 2.70e27·14-s + 2.22e27·15-s + 2.47e28·16-s − 1.02e29·17-s − 3.45e28·18-s + 1.02e30·19-s + 3.83e30·20-s + 1.96e31·21-s − 1.92e31·22-s + 8.76e30·23-s − 2.88e32·24-s − 7.99e32·25-s + 1.52e33·26-s + 1.92e33·27-s + 3.39e34·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.749·3-s + 3/2·4-s + 0.681·5-s − 1.06·6-s + 2.22·7-s − 1.41·8-s + 0.0775·9-s − 0.964·10-s + 0.386·11-s + 1.12·12-s − 0.605·13-s − 3.14·14-s + 0.511·15-s + 5/4·16-s − 1.24·17-s − 0.109·18-s + 0.908·19-s + 1.02·20-s + 1.66·21-s − 0.546·22-s + 0.0874·23-s − 1.06·24-s − 1.12·25-s + 0.856·26-s + 0.444·27-s + 3.33·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(48-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+47/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $1$
Analytic conductor: \(782.966\)
Root analytic conductor: \(5.28975\)
Motivic weight: \(47\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4,\ (\ :47/2, 47/2),\ 1)\)

Particular Values

\(L(24)\) \(\approx\) \(3.523873109\)
\(L(\frac12)\) \(\approx\) \(3.523873109\)
\(L(\frac{49}{2})\) 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)$
bad2$C_1$ \( ( 1 + p^{23} T )^{2} \)
good3$D_{4}$ \( 1 - 4529253512 p^{3} T + 218342028119393782 p^{10} T^{2} - 4529253512 p^{50} T^{3} + p^{94} T^{4} \)
5$D_{4}$ \( 1 - 3635699567838828 p T + \)\(14\!\cdots\!54\)\( p^{7} T^{2} - 3635699567838828 p^{48} T^{3} + p^{94} T^{4} \)
7$D_{4}$ \( 1 - 3286092764408710768 p^{2} T + \)\(19\!\cdots\!94\)\( p^{7} T^{2} - 3286092764408710768 p^{49} T^{3} + p^{94} T^{4} \)
11$D_{4}$ \( 1 - \)\(11\!\cdots\!84\)\( T + \)\(50\!\cdots\!26\)\( p^{3} T^{2} - \)\(11\!\cdots\!84\)\( p^{47} T^{3} + p^{94} T^{4} \)
13$D_{4}$ \( 1 + \)\(70\!\cdots\!72\)\( p T + \)\(42\!\cdots\!78\)\( p^{4} T^{2} + \)\(70\!\cdots\!72\)\( p^{48} T^{3} + p^{94} T^{4} \)
17$D_{4}$ \( 1 + \)\(10\!\cdots\!88\)\( T + \)\(45\!\cdots\!46\)\( p T^{2} + \)\(10\!\cdots\!88\)\( p^{47} T^{3} + p^{94} T^{4} \)
19$D_{4}$ \( 1 - \)\(53\!\cdots\!80\)\( p T + \)\(34\!\cdots\!42\)\( p^{3} T^{2} - \)\(53\!\cdots\!80\)\( p^{48} T^{3} + p^{94} T^{4} \)
23$D_{4}$ \( 1 - \)\(16\!\cdots\!56\)\( p^{2} T + \)\(36\!\cdots\!22\)\( p^{2} T^{2} - \)\(16\!\cdots\!56\)\( p^{49} T^{3} + p^{94} T^{4} \)
29$D_{4}$ \( 1 - \)\(44\!\cdots\!60\)\( T + \)\(13\!\cdots\!98\)\( p^{2} T^{2} - \)\(44\!\cdots\!60\)\( p^{47} T^{3} + p^{94} T^{4} \)
31$D_{4}$ \( 1 - \)\(66\!\cdots\!84\)\( p T + \)\(32\!\cdots\!66\)\( p^{2} T^{2} - \)\(66\!\cdots\!84\)\( p^{48} T^{3} + p^{94} T^{4} \)
37$D_{4}$ \( 1 - \)\(41\!\cdots\!96\)\( p T + \)\(11\!\cdots\!18\)\( p^{2} T^{2} - \)\(41\!\cdots\!96\)\( p^{48} T^{3} + p^{94} T^{4} \)
41$D_{4}$ \( 1 + \)\(18\!\cdots\!16\)\( T + \)\(51\!\cdots\!86\)\( p T^{2} + \)\(18\!\cdots\!16\)\( p^{47} T^{3} + p^{94} T^{4} \)
43$D_{4}$ \( 1 - \)\(58\!\cdots\!28\)\( p T + \)\(40\!\cdots\!82\)\( p^{2} T^{2} - \)\(58\!\cdots\!28\)\( p^{48} T^{3} + p^{94} T^{4} \)
47$D_{4}$ \( 1 - \)\(28\!\cdots\!12\)\( T + \)\(55\!\cdots\!62\)\( T^{2} - \)\(28\!\cdots\!12\)\( p^{47} T^{3} + p^{94} T^{4} \)
53$D_{4}$ \( 1 - \)\(36\!\cdots\!64\)\( T + \)\(24\!\cdots\!98\)\( T^{2} - \)\(36\!\cdots\!64\)\( p^{47} T^{3} + p^{94} T^{4} \)
59$D_{4}$ \( 1 - \)\(21\!\cdots\!20\)\( T + \)\(11\!\cdots\!38\)\( T^{2} - \)\(21\!\cdots\!20\)\( p^{47} T^{3} + p^{94} T^{4} \)
61$D_{4}$ \( 1 - \)\(23\!\cdots\!04\)\( T + \)\(29\!\cdots\!46\)\( T^{2} - \)\(23\!\cdots\!04\)\( p^{47} T^{3} + p^{94} T^{4} \)
67$D_{4}$ \( 1 - \)\(96\!\cdots\!92\)\( T + \)\(11\!\cdots\!62\)\( T^{2} - \)\(96\!\cdots\!92\)\( p^{47} T^{3} + p^{94} T^{4} \)
71$D_{4}$ \( 1 + \)\(34\!\cdots\!56\)\( T + \)\(23\!\cdots\!66\)\( T^{2} + \)\(34\!\cdots\!56\)\( p^{47} T^{3} + p^{94} T^{4} \)
73$D_{4}$ \( 1 + \)\(44\!\cdots\!36\)\( T + \)\(77\!\cdots\!18\)\( T^{2} + \)\(44\!\cdots\!36\)\( p^{47} T^{3} + p^{94} T^{4} \)
79$D_{4}$ \( 1 - \)\(45\!\cdots\!00\)\( T + \)\(19\!\cdots\!18\)\( T^{2} - \)\(45\!\cdots\!00\)\( p^{47} T^{3} + p^{94} T^{4} \)
83$D_{4}$ \( 1 + \)\(13\!\cdots\!16\)\( T + \)\(73\!\cdots\!18\)\( T^{2} + \)\(13\!\cdots\!16\)\( p^{47} T^{3} + p^{94} T^{4} \)
89$D_{4}$ \( 1 - \)\(45\!\cdots\!20\)\( T + \)\(87\!\cdots\!58\)\( T^{2} - \)\(45\!\cdots\!20\)\( p^{47} T^{3} + p^{94} T^{4} \)
97$D_{4}$ \( 1 - \)\(53\!\cdots\!92\)\( T + \)\(44\!\cdots\!42\)\( T^{2} - \)\(53\!\cdots\!92\)\( p^{47} T^{3} + p^{94} T^{4} \)
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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

−17.57737253822962340094157557889, −17.30869601008150832221953436288, −15.89473386084642124104296984959, −15.03254116980632639876689598475, −14.27660402495919206933351349254, −13.62800413382491004312772591449, −11.81217559882232275802010381244, −11.49467038133457401683473219702, −10.33106861661974574366037375633, −9.661025476722952792623206304555, −8.513817036919953016245823537554, −8.349471305303799656929193553936, −7.42408072172744007028117126382, −6.42932931965470428473469848105, −5.17161289649930107787005837040, −4.28534652448586902058129702984, −2.52476585088168058541224491049, −2.31698083471204996760112830146, −1.30729127808266273151362806314, −0.816104697295273947138718633283, 0.816104697295273947138718633283, 1.30729127808266273151362806314, 2.31698083471204996760112830146, 2.52476585088168058541224491049, 4.28534652448586902058129702984, 5.17161289649930107787005837040, 6.42932931965470428473469848105, 7.42408072172744007028117126382, 8.349471305303799656929193553936, 8.513817036919953016245823537554, 9.661025476722952792623206304555, 10.33106861661974574366037375633, 11.49467038133457401683473219702, 11.81217559882232275802010381244, 13.62800413382491004312772591449, 14.27660402495919206933351349254, 15.03254116980632639876689598475, 15.89473386084642124104296984959, 17.30869601008150832221953436288, 17.57737253822962340094157557889

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