Properties

Label 4-1216e2-1.1-c2e2-0-5
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 + 22/7·7-s − 1.55·9-s − 0.545·11-s − 2·17-s + 2·19-s + 4/23·23-s + 3.87·25-s − 8.79·35-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 − 4.88·63-s + 1.06·73-s − 1.71·77-s + 1.41·81-s + 0.144·83-s + 28/5·85-s − 5.59·95-s + 0.848·99-s − 2.41·101-s − 0.486·115-s − 6.28·119-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

\(L(\frac{3}{2})\) \(\approx\) \(1.066889051\)
\(L(\frac12)\) \(\approx\) \(1.066889051\)
\(L(2)\) 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

−9.540650767681941359692805003669, −9.062383584833459763181741976473, −8.696617820803290269288328945828, −8.299996811489996487599868109139, −8.069110672223329634107191782531, −7.892922880357853441373294662679, −7.48203918408387780751839318903, −7.19874399972789329117563402456, −6.61638475281878046062824794978, −5.71792507495550702804150762572, −5.29699890288535715207902447001, −5.00628248639374506903597141782, −4.47404142414167805566248661023, −4.30850473646962869543015857749, −3.74205890719660940708967061462, −3.09913168100372625707754702975, −2.58633157320536992599630186008, −1.89600735500105100586919915613, −1.04741419332535187604488460019, −0.36222668978196687447089528847, 0.36222668978196687447089528847, 1.04741419332535187604488460019, 1.89600735500105100586919915613, 2.58633157320536992599630186008, 3.09913168100372625707754702975, 3.74205890719660940708967061462, 4.30850473646962869543015857749, 4.47404142414167805566248661023, 5.00628248639374506903597141782, 5.29699890288535715207902447001, 5.71792507495550702804150762572, 6.61638475281878046062824794978, 7.19874399972789329117563402456, 7.48203918408387780751839318903, 7.892922880357853441373294662679, 8.069110672223329634107191782531, 8.299996811489996487599868109139, 8.696617820803290269288328945828, 9.062383584833459763181741976473, 9.540650767681941359692805003669

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