Properties

Label 4-1216e2-1.1-c2e2-0-2
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $1097.83$
Root an. cond. $5.75617$
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  + 2·5-s + 10·7-s + 10·9-s − 10·11-s − 50·17-s − 38·19-s − 20·23-s − 47·25-s + 20·35-s − 10·43-s + 20·45-s + 10·47-s − 23·49-s − 20·55-s − 190·61-s + 100·63-s − 50·73-s − 100·77-s + 19·81-s + 260·83-s − 100·85-s − 76·95-s − 100·99-s − 40·115-s − 500·119-s − 167·121-s − 146·125-s + ⋯
L(s)  = 1  + 2/5·5-s + 10/7·7-s + 10/9·9-s − 0.909·11-s − 2.94·17-s − 2·19-s − 0.869·23-s − 1.87·25-s + 4/7·35-s − 0.232·43-s + 4/9·45-s + 0.212·47-s − 0.469·49-s − 0.363·55-s − 3.11·61-s + 1.58·63-s − 0.684·73-s − 1.29·77-s + 0.234·81-s + 3.13·83-s − 1.17·85-s − 4/5·95-s − 1.01·99-s − 0.347·115-s − 4.20·119-s − 1.38·121-s − 1.16·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1097.83\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5770402607\)
\(L(\frac12)\) \(\approx\) \(0.5770402607\)
\(L(2)\) 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)$
bad2 \( 1 \)
19$C_1$ \( ( 1 + p T )^{2} \)
good3$C_2^2$ \( 1 - 10 T^{2} + p^{4} T^{4} \)
5$C_2$ \( ( 1 - T + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 - 5 T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 + 5 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 50 T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 + 25 T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
29$C_2^2$ \( 1 + 118 T^{2} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 122 T^{2} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 2090 T^{2} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 1562 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 5 T + p^{2} T^{2} )^{2} \)
47$C_2$ \( ( 1 - 5 T + p^{2} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 4970 T^{2} + p^{4} T^{4} \)
59$C_2$ \( ( 1 - 82 T + p^{2} T^{2} )( 1 + 82 T + p^{2} T^{2} ) \)
61$C_2$ \( ( 1 + 95 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 3190 T^{2} + p^{4} T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + 25 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 10682 T^{2} + p^{4} T^{4} \)
83$C_2$ \( ( 1 - 130 T + p^{2} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 358 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 18530 T^{2} + p^{4} T^{4} \)
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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

−10.14211977750432081698634536136, −9.111125424848182878019287474504, −9.064956955674974362968109550334, −8.467560891220189152288844627721, −8.110173391343305682203300008268, −7.61380158389470212511342397410, −7.58872154355642077549077883763, −6.59343707111998454952409736547, −6.53208990903810547201592619515, −6.12695470832555935902803528191, −5.49169899014988758276024174389, −4.80668925249309729935927876048, −4.66637375942030275674107556850, −4.10572393155576252417625511179, −4.03002667762913042496175513442, −2.88318430699156944380576109765, −2.11644070791355037916296215588, −1.96391995665077893129981685200, −1.63840961604566901259155954811, −0.20304723731641667824912255801, 0.20304723731641667824912255801, 1.63840961604566901259155954811, 1.96391995665077893129981685200, 2.11644070791355037916296215588, 2.88318430699156944380576109765, 4.03002667762913042496175513442, 4.10572393155576252417625511179, 4.66637375942030275674107556850, 4.80668925249309729935927876048, 5.49169899014988758276024174389, 6.12695470832555935902803528191, 6.53208990903810547201592619515, 6.59343707111998454952409736547, 7.58872154355642077549077883763, 7.61380158389470212511342397410, 8.110173391343305682203300008268, 8.467560891220189152288844627721, 9.064956955674974362968109550334, 9.111125424848182878019287474504, 10.14211977750432081698634536136

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