Properties

Label 4-76e2-1.1-c4e2-0-0
Degree $4$
Conductor $5776$
Sign $1$
Analytic cond. $61.7185$
Root an. cond. $2.80287$
Motivic weight $4$
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  − 31·5-s + 73·7-s + 162·9-s + 233·11-s + 353·17-s + 722·19-s − 316·23-s + 625·25-s − 2.26e3·35-s − 3.52e3·43-s − 5.02e3·45-s − 1.20e3·47-s + 2.40e3·49-s − 7.22e3·55-s − 3.16e3·61-s + 1.18e4·63-s + 1.00e4·73-s + 1.70e4·77-s + 1.96e4·81-s − 1.13e4·83-s − 1.09e4·85-s − 2.23e4·95-s + 3.77e4·99-s − 1.99e4·101-s + 9.79e3·115-s + 2.57e4·119-s + 1.46e4·121-s + ⋯
L(s)  = 1  − 1.23·5-s + 1.48·7-s + 2·9-s + 1.92·11-s + 1.22·17-s + 2·19-s − 0.597·23-s + 25-s − 1.84·35-s − 1.90·43-s − 2.47·45-s − 0.546·47-s + 49-s − 2.38·55-s − 0.851·61-s + 2.97·63-s + 1.88·73-s + 2.86·77-s + 3·81-s − 1.64·83-s − 1.51·85-s − 2.47·95-s + 3.85·99-s − 1.96·101-s + 0.740·115-s + 1.81·119-s + 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5776\)    =    \(2^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(61.7185\)
Root analytic conductor: \(2.80287\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{76} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5776,\ (\ :2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.157129682\)
\(L(\frac12)\) \(\approx\) \(3.157129682\)
\(L(3)\) 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$C_1$ \( ( 1 - p^{2} T )^{2} \)
good3$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
5$C_2^2$ \( 1 + 31 T + 336 T^{2} + 31 p^{4} T^{3} + p^{8} T^{4} \)
7$C_2^2$ \( 1 - 73 T + 2928 T^{2} - 73 p^{4} T^{3} + p^{8} T^{4} \)
11$C_2^2$ \( 1 - 233 T + 39648 T^{2} - 233 p^{4} T^{3} + p^{8} T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
17$C_2^2$ \( 1 - 353 T + 41088 T^{2} - 353 p^{4} T^{3} + p^{8} T^{4} \)
23$C_2$ \( ( 1 + 158 T + p^{4} T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
43$C_2^2$ \( 1 + 3527 T + 9020928 T^{2} + 3527 p^{4} T^{3} + p^{8} T^{4} \)
47$C_2^2$ \( 1 + 1207 T - 3422832 T^{2} + 1207 p^{4} T^{3} + p^{8} T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
61$C_2^2$ \( 1 + 3167 T - 3815952 T^{2} + 3167 p^{4} T^{3} + p^{8} T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
73$C_2^2$ \( 1 - 10033 T + 72262848 T^{2} - 10033 p^{4} T^{3} + p^{8} T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
83$C_2$ \( ( 1 + 5678 T + p^{4} T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
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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

−14.01904700270578621902468101096, −13.74744562895246542081352818785, −12.74095326069789419674494380532, −12.21847304437388788078895394684, −11.70864465853710102278416160415, −11.64468282297172972930385943609, −10.85774403611172679270625016559, −9.981556671109526718291112115944, −9.682769044989625067952066346482, −8.918084673256899788358037017479, −7.894755410937480218167634326307, −7.87035126533830974706040219243, −7.09606906488378120870053667561, −6.60156746884170893242163577892, −5.29732262375819872334377315942, −4.67789452048106077389420418400, −3.96023389592170013474671415225, −3.49104535304896258821772151362, −1.42783578818619358528327281108, −1.22965593474006071425227239528, 1.22965593474006071425227239528, 1.42783578818619358528327281108, 3.49104535304896258821772151362, 3.96023389592170013474671415225, 4.67789452048106077389420418400, 5.29732262375819872334377315942, 6.60156746884170893242163577892, 7.09606906488378120870053667561, 7.87035126533830974706040219243, 7.894755410937480218167634326307, 8.918084673256899788358037017479, 9.682769044989625067952066346482, 9.981556671109526718291112115944, 10.85774403611172679270625016559, 11.64468282297172972930385943609, 11.70864465853710102278416160415, 12.21847304437388788078895394684, 12.74095326069789419674494380532, 13.74744562895246542081352818785, 14.01904700270578621902468101096

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