Properties

Label 4-768e2-1.1-c1e2-0-0
Degree $4$
Conductor $589824$
Sign $1$
Analytic cond. $37.6076$
Root an. cond. $2.47639$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 9-s − 4·17-s − 8·23-s − 6·25-s − 12·31-s − 12·41-s + 8·47-s − 2·49-s + 4·63-s − 24·71-s + 20·73-s + 20·79-s + 81-s + 28·89-s + 20·97-s + 20·103-s + 4·113-s + 16·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + ⋯
L(s)  = 1  − 1.51·7-s − 1/3·9-s − 0.970·17-s − 1.66·23-s − 6/5·25-s − 2.15·31-s − 1.87·41-s + 1.16·47-s − 2/7·49-s + 0.503·63-s − 2.84·71-s + 2.34·73-s + 2.25·79-s + 1/9·81-s + 2.96·89-s + 2.03·97-s + 1.97·103-s + 0.376·113-s + 1.46·119-s + 6/11·121-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 + 0.323·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(589824\)    =    \(2^{16} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(37.6076\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 589824,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4793919349\)
\(L(\frac12)\) \(\approx\) \(0.4793919349\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−10.57008215204519927246407898417, −10.05730450095804230643946501346, −9.653336690991456243487096429627, −9.322670663521917508457169679837, −8.894496537925992005346043078937, −8.542818380318845554381638819095, −7.86833136205618990045713220876, −7.53158469482663380559311925234, −7.08897545804253260832650456486, −6.38838432454398027286300449672, −6.23306808906560510342538970748, −5.87826601277013489916536709141, −5.15622401048495279540220406500, −4.72891855256181988606217508400, −3.79702272776298811740541809360, −3.68246780667453146020142799097, −3.17006555083538106967977607268, −2.10181313392884375724421942125, −2.00540464051609260910892227583, −0.32769642696164508104003008851, 0.32769642696164508104003008851, 2.00540464051609260910892227583, 2.10181313392884375724421942125, 3.17006555083538106967977607268, 3.68246780667453146020142799097, 3.79702272776298811740541809360, 4.72891855256181988606217508400, 5.15622401048495279540220406500, 5.87826601277013489916536709141, 6.23306808906560510342538970748, 6.38838432454398027286300449672, 7.08897545804253260832650456486, 7.53158469482663380559311925234, 7.86833136205618990045713220876, 8.542818380318845554381638819095, 8.894496537925992005346043078937, 9.322670663521917508457169679837, 9.653336690991456243487096429627, 10.05730450095804230643946501346, 10.57008215204519927246407898417

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