Properties

Label 4-1216e2-1.1-c2e2-0-8
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  − 8·5-s − 10·7-s + 5·9-s + 20·11-s + 30·17-s + 12·19-s + 70·23-s − 2·25-s + 80·35-s + 40·43-s − 40·45-s + 20·47-s − 23·49-s − 160·55-s + 80·61-s − 50·63-s + 210·73-s − 200·77-s − 56·81-s + 80·83-s − 240·85-s − 96·95-s + 100·99-s − 560·115-s − 300·119-s + 58·121-s + 344·125-s + ⋯
L(s)  = 1  − 8/5·5-s − 1.42·7-s + 5/9·9-s + 1.81·11-s + 1.76·17-s + 0.631·19-s + 3.04·23-s − 0.0799·25-s + 16/7·35-s + 0.930·43-s − 8/9·45-s + 0.425·47-s − 0.469·49-s − 2.90·55-s + 1.31·61-s − 0.793·63-s + 2.87·73-s − 2.59·77-s − 0.691·81-s + 0.963·83-s − 2.82·85-s − 1.01·95-s + 1.01·99-s − 4.86·115-s − 2.52·119-s + 0.479·121-s + 2.75·125-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

\(L(\frac{3}{2})\) \(\approx\) \(2.612875078\)
\(L(\frac12)\) \(\approx\) \(2.612875078\)
\(L(2)\) 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$C_2$ \( 1 - 12 T + p^{2} T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{4} T^{4} \)
5$C_2$ \( ( 1 + 4 T + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 + 5 T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 p T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 - 15 T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 - 35 T + p^{2} T^{2} )^{2} \)
29$C_2^2$ \( 1 - 1357 T^{2} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 622 T^{2} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 2270 T^{2} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 2062 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 20 T + p^{2} T^{2} )^{2} \)
47$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
53$C_2^2$ \( 1 + 115 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 6637 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 7405 T^{2} + p^{4} T^{4} \)
71$C_2$ \( ( 1 - 92 T + p^{2} T^{2} )( 1 + 92 T + p^{2} T^{2} ) \)
73$C_2$ \( ( 1 - 105 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 11182 T^{2} + p^{4} T^{4} \)
83$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_2^2$ \( 1 - 3790 T^{2} + p^{4} T^{4} \)
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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

−9.438895029596018996698915995542, −9.353978421392044473055235757074, −9.259672820298546694606943786140, −8.426679643083524346478007770617, −8.133409426808966217802246352690, −7.54139763944155115374390247922, −7.26823967131960976885305625610, −6.99828808117022143006885239206, −6.36113770135530673362095387434, −6.32926485794604090255931028139, −5.27158224752882950071710793233, −5.26431979246772783795634852413, −4.37945596109797372822628462439, −3.90011732314321669000611336938, −3.66430254819200043153243228548, −3.19821733728590794944822983612, −2.90351327032917068319421038687, −1.71638275745860960718134510892, −0.849231700049078371256387400613, −0.71927980831367581089785655597, 0.71927980831367581089785655597, 0.849231700049078371256387400613, 1.71638275745860960718134510892, 2.90351327032917068319421038687, 3.19821733728590794944822983612, 3.66430254819200043153243228548, 3.90011732314321669000611336938, 4.37945596109797372822628462439, 5.26431979246772783795634852413, 5.27158224752882950071710793233, 6.32926485794604090255931028139, 6.36113770135530673362095387434, 6.99828808117022143006885239206, 7.26823967131960976885305625610, 7.54139763944155115374390247922, 8.133409426808966217802246352690, 8.426679643083524346478007770617, 9.259672820298546694606943786140, 9.353978421392044473055235757074, 9.438895029596018996698915995542

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