Properties

Label 4-1216e2-1.1-c2e2-0-0
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  − 14·5-s − 22·7-s − 14·9-s + 6·11-s − 34·17-s − 38·19-s − 4·23-s + 97·25-s + 308·35-s − 42·43-s + 196·45-s + 10·47-s + 265·49-s − 84·55-s − 46·61-s + 308·63-s + 78·73-s − 132·77-s + 115·81-s − 12·83-s + 476·85-s + 532·95-s − 84·99-s − 244·101-s + 56·115-s + 748·119-s − 215·121-s + ⋯
L(s)  = 1  − 2.79·5-s − 3.14·7-s − 1.55·9-s + 6/11·11-s − 2·17-s − 2·19-s − 0.173·23-s + 3.87·25-s + 44/5·35-s − 0.976·43-s + 4.35·45-s + 0.212·47-s + 5.40·49-s − 1.52·55-s − 0.754·61-s + 44/9·63-s + 1.06·73-s − 1.71·77-s + 1.41·81-s − 0.144·83-s + 28/5·85-s + 28/5·95-s − 0.848·99-s − 2.41·101-s + 0.486·115-s + 44/7·119-s − 1.77·121-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

\(L(\frac{3}{2})\) \(\approx\) \(0.004613392093\)
\(L(\frac12)\) \(\approx\) \(0.004613392093\)
\(L(2)\) 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$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \)
5$C_2$ \( ( 1 + 7 T + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 + 11 T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 210 T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{2} \)
29$C_2^2$ \( 1 - 114 T^{2} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 1890 T^{2} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 1170 T^{2} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 1794 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 21 T + p^{2} T^{2} )^{2} \)
47$C_2$ \( ( 1 - 5 T + p^{2} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 5586 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 5810 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 23 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 7410 T^{2} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 1890 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 39 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 3234 T^{2} + p^{4} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 1730 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 - 94 T + p^{2} T^{2} )( 1 + 94 T + p^{2} T^{2} ) \)
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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

−10.31150705842521774838538054939, −9.499504841946765614837880419047, −8.938076511698764801202415780583, −8.431825644146934926656598884659, −8.277787125946209191810643769546, −7.80743403561272948359982110156, −7.02308606120827680893592290552, −6.87427926301022186856762226721, −6.50506623033683166689465842016, −6.24383336522533258810506020173, −5.74116235583849464221598370865, −4.89053807554149028884187376236, −4.16865018399797391887960700234, −4.08562084291334047450648913169, −3.64239698037817293357079682601, −3.15777091138946962357321195091, −2.82158798408060234573502600703, −2.21794971763085243798823570837, −0.48773837777327157574429027387, −0.04288121501771759171089913368, 0.04288121501771759171089913368, 0.48773837777327157574429027387, 2.21794971763085243798823570837, 2.82158798408060234573502600703, 3.15777091138946962357321195091, 3.64239698037817293357079682601, 4.08562084291334047450648913169, 4.16865018399797391887960700234, 4.89053807554149028884187376236, 5.74116235583849464221598370865, 6.24383336522533258810506020173, 6.50506623033683166689465842016, 6.87427926301022186856762226721, 7.02308606120827680893592290552, 7.80743403561272948359982110156, 8.277787125946209191810643769546, 8.431825644146934926656598884659, 8.938076511698764801202415780583, 9.499504841946765614837880419047, 10.31150705842521774838538054939

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