Properties

Label 4-3724e2-1.1-c0e2-0-0
Degree $4$
Conductor $13868176$
Sign $1$
Analytic cond. $3.45408$
Root an. cond. $1.36327$
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  − 5-s − 9-s + 11-s − 17-s + 19-s − 2·23-s + 25-s − 2·43-s + 45-s − 47-s − 55-s − 61-s − 73-s − 4·83-s + 85-s − 95-s − 99-s + 2·101-s + 2·115-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 5-s − 9-s + 11-s − 17-s + 19-s − 2·23-s + 25-s − 2·43-s + 45-s − 47-s − 55-s − 61-s − 73-s − 4·83-s + 85-s − 95-s − 99-s + 2·101-s + 2·115-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13868176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13868176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13868176\)    =    \(2^{4} \cdot 7^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(3.45408\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13868176,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6255699894\)
\(L(\frac12)\) \(\approx\) \(0.6255699894\)
\(L(1)\) 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 \( 1 \)
19$C_2$ \( 1 - T + T^{2} \)
good3$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
23$C_2$ \( ( 1 + T + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T + T^{2} )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_1$ \( ( 1 + T )^{4} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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

−9.048616211462397993847955943489, −8.484499916256035092218116157487, −8.057432989909275422487080002864, −8.030873348295406610039220135554, −7.35778004714409328051748199250, −7.04136499346802271328214882555, −6.76694191394180329297765370560, −6.20290676487627219566036145459, −6.04244948035794181222370909854, −5.52147903420193745335290785175, −5.14319191255763356976977400963, −4.44912547678862069342612598912, −4.42941452020313430966528628575, −3.87516854703353057433343640847, −3.42604861988246881576847567162, −3.02763530166876607375185831068, −2.69553600530230036484334596130, −1.69620310924035666304499455531, −1.66573020750314684042702900610, −0.44657338964641443807477173670, 0.44657338964641443807477173670, 1.66573020750314684042702900610, 1.69620310924035666304499455531, 2.69553600530230036484334596130, 3.02763530166876607375185831068, 3.42604861988246881576847567162, 3.87516854703353057433343640847, 4.42941452020313430966528628575, 4.44912547678862069342612598912, 5.14319191255763356976977400963, 5.52147903420193745335290785175, 6.04244948035794181222370909854, 6.20290676487627219566036145459, 6.76694191394180329297765370560, 7.04136499346802271328214882555, 7.35778004714409328051748199250, 8.030873348295406610039220135554, 8.057432989909275422487080002864, 8.484499916256035092218116157487, 9.048616211462397993847955943489

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