Properties

Label 4-1216e2-1.1-c2e2-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·5-s − 2·7-s − 11·9-s − 28·11-s + 46·17-s − 20·19-s − 2·23-s − 2·25-s − 16·35-s − 136·43-s − 88·45-s + 52·47-s − 95·49-s − 224·55-s + 80·61-s + 22·63-s − 14·73-s + 56·77-s + 40·81-s − 64·83-s + 368·85-s − 160·95-s + 308·99-s − 28·101-s − 16·115-s − 92·119-s + 346·121-s + ⋯
L(s)  = 1  + 8/5·5-s − 2/7·7-s − 1.22·9-s − 2.54·11-s + 2.70·17-s − 1.05·19-s − 0.0869·23-s − 0.0799·25-s − 0.457·35-s − 3.16·43-s − 1.95·45-s + 1.10·47-s − 1.93·49-s − 4.07·55-s + 1.31·61-s + 0.349·63-s − 0.191·73-s + 8/11·77-s + 0.493·81-s − 0.771·83-s + 4.32·85-s − 1.68·95-s + 28/9·99-s − 0.277·101-s − 0.139·115-s − 0.773·119-s + 2.85·121-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

\(L(\frac{3}{2})\) \(\approx\) \(0.7153281381\)
\(L(\frac12)\) \(\approx\) \(0.7153281381\)
\(L(2)\) 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$C_2$ \( 1 + 20 T + p^{2} T^{2} \)
good3$C_2^2$ \( 1 + 11 T^{2} + p^{4} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 + 14 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 77 T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 - 23 T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
29$C_2^2$ \( 1 + 23 p T^{2} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 878 T^{2} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 1694 T^{2} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 2318 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 68 T + p^{2} T^{2} )^{2} \)
47$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{2} \)
53$C_2^2$ \( 1 + 907 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 6701 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 8717 T^{2} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 9038 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 7 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 3086 T^{2} + p^{4} T^{4} \)
83$C_2$ \( ( 1 + 32 T + p^{2} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 862 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 9422 T^{2} + p^{4} T^{4} \)
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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

−9.957842250745453315085466492921, −9.686034297205102134813149676255, −8.746110701667707452973170001986, −8.623935201419887579924836794626, −7.939612379152668871728173979335, −7.904138039466684578776350792751, −7.49862133203589999474035802388, −6.51943767482147746703159862913, −6.47517827988145714587391395097, −5.69626860297805795168906230604, −5.59632782012775843405765928241, −5.22723324666964241229507480778, −5.03104254465385912714353106638, −4.05437807029188097379840592267, −3.19141227860946035444838793409, −3.14207803522293706239999819231, −2.46789956825508494439354321580, −2.03788015446159387584131185698, −1.38092686983196386516159175989, −0.23552244140234546764903582659, 0.23552244140234546764903582659, 1.38092686983196386516159175989, 2.03788015446159387584131185698, 2.46789956825508494439354321580, 3.14207803522293706239999819231, 3.19141227860946035444838793409, 4.05437807029188097379840592267, 5.03104254465385912714353106638, 5.22723324666964241229507480778, 5.59632782012775843405765928241, 5.69626860297805795168906230604, 6.47517827988145714587391395097, 6.51943767482147746703159862913, 7.49862133203589999474035802388, 7.904138039466684578776350792751, 7.939612379152668871728173979335, 8.623935201419887579924836794626, 8.746110701667707452973170001986, 9.686034297205102134813149676255, 9.957842250745453315085466492921

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