Properties

Label 4-76e2-1.1-c3e2-0-0
Degree $4$
Conductor $5776$
Sign $1$
Analytic cond. $20.1075$
Root an. cond. $2.11758$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·3-s − 5·5-s − 30·7-s − 27·9-s − 71·11-s − 35·13-s + 25·15-s − 38·19-s + 150·21-s − 5·23-s − 25·25-s + 260·27-s + 155·29-s − 88·31-s + 355·33-s + 150·35-s + 380·37-s + 175·39-s − 142·41-s + 155·43-s + 135·45-s − 455·47-s + 121·49-s − 275·53-s + 355·55-s + 190·57-s − 873·59-s + ⋯
L(s)  = 1  − 0.962·3-s − 0.447·5-s − 1.61·7-s − 9-s − 1.94·11-s − 0.746·13-s + 0.430·15-s − 0.458·19-s + 1.55·21-s − 0.0453·23-s − 1/5·25-s + 1.85·27-s + 0.992·29-s − 0.509·31-s + 1.87·33-s + 0.724·35-s + 1.68·37-s + 0.718·39-s − 0.540·41-s + 0.549·43-s + 0.447·45-s − 1.41·47-s + 0.352·49-s − 0.712·53-s + 0.870·55-s + 0.441·57-s − 1.92·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5776\)    =    \(2^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(20.1075\)
Root analytic conductor: \(2.11758\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{76} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5776,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 + p T )^{2} \)
good3$D_{4}$ \( 1 + 5 T + 52 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 + p T + 2 p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 30 T + 779 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 71 T + 3716 T^{2} + 71 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 35 T + 3702 T^{2} + 35 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 9529 T^{2} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 5 T - 4378 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 155 T + 19928 T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 88 T + 8718 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 380 T + 136878 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 142 T + 135458 T^{2} + 142 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 155 T + 52086 T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 455 T + 255038 T^{2} + 455 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 275 T + 191846 T^{2} + 275 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 873 T + 591184 T^{2} + 873 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 445 T + 250812 T^{2} - 445 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 645 T + 674834 T^{2} - 645 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1712 T + 1445258 T^{2} + 1712 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 990 T + 989267 T^{2} + 990 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 1274 T + 1391022 T^{2} + 1274 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 90 T + 1038382 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 888 T + 1554274 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 710 T + 220818 T^{2} - 710 p^{3} T^{3} + p^{6} T^{4} \)
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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

−13.49788503722776660302967981564, −12.99159099534972308548603158684, −12.75174266183647683931669042549, −12.11275575634791519633455987561, −11.48177925080151079000655296282, −11.10944874209083542019704076214, −10.24642379119172767704449692881, −10.13738636923699982955297322701, −9.289819188237190567974266006637, −8.511644614083163592386551975397, −7.896494004176568198328012953839, −7.24869569877946835792619065648, −6.30155090606266999785009557310, −6.00316224674392827034502495593, −5.21601495993563621097175110718, −4.52575892854559574122108446838, −3.07106717455804559315000004384, −2.74949077236954508699012559256, 0, 0, 2.74949077236954508699012559256, 3.07106717455804559315000004384, 4.52575892854559574122108446838, 5.21601495993563621097175110718, 6.00316224674392827034502495593, 6.30155090606266999785009557310, 7.24869569877946835792619065648, 7.896494004176568198328012953839, 8.511644614083163592386551975397, 9.289819188237190567974266006637, 10.13738636923699982955297322701, 10.24642379119172767704449692881, 11.10944874209083542019704076214, 11.48177925080151079000655296282, 12.11275575634791519633455987561, 12.75174266183647683931669042549, 12.99159099534972308548603158684, 13.49788503722776660302967981564

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