Properties

Label 4-4608e2-1.1-c1e2-0-33
Degree $4$
Conductor $21233664$
Sign $1$
Analytic cond. $1353.87$
Root an. cond. $6.06589$
Motivic weight $1$
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  + 8·7-s + 8·17-s − 8·23-s + 2·25-s − 16·31-s + 4·41-s + 34·49-s + 8·71-s + 8·73-s + 16·79-s + 24·89-s − 8·97-s + 24·103-s − 28·113-s + 64·119-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 64·161-s + 163-s + 167-s + 18·169-s + ⋯
L(s)  = 1  + 3.02·7-s + 1.94·17-s − 1.66·23-s + 2/5·25-s − 2.87·31-s + 0.624·41-s + 34/7·49-s + 0.949·71-s + 0.936·73-s + 1.80·79-s + 2.54·89-s − 0.812·97-s + 2.36·103-s − 2.63·113-s + 5.86·119-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 5.04·161-s + 0.0783·163-s + 0.0773·167-s + 1.38·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21233664\)    =    \(2^{18} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1353.87\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4608} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21233664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.874413793\)
\(L(\frac12)\) \(\approx\) \(4.874413793\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 100 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 116 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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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

−8.161048167499630293320706591908, −8.033404785342362874929681190587, −7.85707408947901313955270006254, −7.69053922091741220420945393274, −7.24084011673961309058733793923, −6.78719626837114206837172959974, −6.23759932069706447639389250404, −5.65824006542827528276795866023, −5.59278205061334039699439425990, −5.11254444484965708242530121742, −4.94967096528404641915530265275, −4.48786736240964925000394711254, −3.90834311019608870711795833723, −3.71205497990449823282899726602, −3.27918955886541564057680195097, −2.33115270495740586752525859178, −2.12474932894076248211621500644, −1.65741733436685305257799002990, −1.27947935886242528408816348254, −0.63473727976544485320925178187, 0.63473727976544485320925178187, 1.27947935886242528408816348254, 1.65741733436685305257799002990, 2.12474932894076248211621500644, 2.33115270495740586752525859178, 3.27918955886541564057680195097, 3.71205497990449823282899726602, 3.90834311019608870711795833723, 4.48786736240964925000394711254, 4.94967096528404641915530265275, 5.11254444484965708242530121742, 5.59278205061334039699439425990, 5.65824006542827528276795866023, 6.23759932069706447639389250404, 6.78719626837114206837172959974, 7.24084011673961309058733793923, 7.69053922091741220420945393274, 7.85707408947901313955270006254, 8.033404785342362874929681190587, 8.161048167499630293320706591908

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