Properties

Label 4-896e2-1.1-c0e2-0-3
Degree $4$
Conductor $802816$
Sign $1$
Analytic cond. $0.199954$
Root an. cond. $0.668701$
Motivic weight $0$
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  + 2·7-s + 3·49-s − 4·71-s − 4·79-s − 81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 2·7-s + 3·49-s − 4·71-s − 4·79-s − 81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(802816\)    =    \(2^{14} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.199954\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{896} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 802816,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.194020124\)
\(L(\frac12)\) \(\approx\) \(1.194020124\)
\(L(1)\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_1$ \( ( 1 + T )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_1$ \( ( 1 + T )^{4} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−10.44660375672113490465912434070, −10.14746318908492613647259307666, −9.914221623344445245420765247528, −8.969697968756809860395976038882, −8.797568559389099666788830059643, −8.596667266846424619982833358831, −7.955671642138657226801464781656, −7.51603439764461457949066151747, −7.38088076323783862002257168116, −6.78870191631647409778969357441, −6.10923970358682529950724072724, −5.59548830859865741659255979477, −5.40223632646380237037549214418, −4.53341792128439175629482969925, −4.51621428394311017800297436680, −3.95256151711235072160277779437, −3.04862959745114902768635689818, −2.57535299837503157275197784454, −1.69662029269896584039931522169, −1.35654336297365492313584603706, 1.35654336297365492313584603706, 1.69662029269896584039931522169, 2.57535299837503157275197784454, 3.04862959745114902768635689818, 3.95256151711235072160277779437, 4.51621428394311017800297436680, 4.53341792128439175629482969925, 5.40223632646380237037549214418, 5.59548830859865741659255979477, 6.10923970358682529950724072724, 6.78870191631647409778969357441, 7.38088076323783862002257168116, 7.51603439764461457949066151747, 7.955671642138657226801464781656, 8.596667266846424619982833358831, 8.797568559389099666788830059643, 8.969697968756809860395976038882, 9.914221623344445245420765247528, 10.14746318908492613647259307666, 10.44660375672113490465912434070

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