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Properties

Label 4-3744e2-1.1-c0e2-0-1
Degree $4$
Conductor $14017536$
Sign $1$
Analytic cond. $3.49129$
Root an. cond. $1.36693$
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  − 2·11-s − 2·13-s − 2·37-s − 2·47-s − 2·59-s − 2·71-s + 2·73-s + 2·83-s + 2·97-s + 4·107-s − 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 4·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 2·11-s − 2·13-s − 2·37-s − 2·47-s − 2·59-s − 2·71-s + 2·73-s + 2·83-s + 2·97-s + 4·107-s − 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 4·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14017536\)    =    \(2^{10} \cdot 3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3.49129\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14017536,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4642242113\)
\(L(\frac12)\) \(\approx\) \(0.4642242113\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
89$C_2^2$ \( 1 + T^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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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

−9.060915216984344029401056407430, −8.479114707396776627801895371896, −7.85884678801776605512345357631, −7.74654126851050169345744502390, −7.67715184209909167718325525339, −7.01610783570122534131349390971, −6.82294545154454196974248832912, −6.24097908528318041254709554502, −5.92636738845667807504939595264, −5.31066003577159365900450029141, −5.00830964200502202475707942690, −4.85934667144354491399694721526, −4.61373105437148096088441184588, −3.71101086828011157979238802297, −3.40484264466651294328229669767, −2.86597792640677678563461773135, −2.54752532615215223820573898486, −2.03471703486190963481334392981, −1.61027865595398589909353528693, −0.36287447309263931697076082015, 0.36287447309263931697076082015, 1.61027865595398589909353528693, 2.03471703486190963481334392981, 2.54752532615215223820573898486, 2.86597792640677678563461773135, 3.40484264466651294328229669767, 3.71101086828011157979238802297, 4.61373105437148096088441184588, 4.85934667144354491399694721526, 5.00830964200502202475707942690, 5.31066003577159365900450029141, 5.92636738845667807504939595264, 6.24097908528318041254709554502, 6.82294545154454196974248832912, 7.01610783570122534131349390971, 7.67715184209909167718325525339, 7.74654126851050169345744502390, 7.85884678801776605512345357631, 8.479114707396776627801895371896, 9.060915216984344029401056407430

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