Properties

Label 4-209088-1.1-c1e2-0-63
Degree $4$
Conductor $209088$
Sign $-1$
Analytic cond. $13.3316$
Root an. cond. $1.91082$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 2·11-s + 12·13-s − 16·23-s − 6·25-s − 27-s + 2·33-s − 12·37-s − 12·39-s − 10·49-s − 24·59-s − 28·61-s + 16·69-s + 12·73-s + 6·75-s + 81-s + 32·83-s − 4·97-s − 2·99-s + 32·107-s − 36·109-s + 12·111-s + 12·117-s + 3·121-s + 127-s + 131-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.603·11-s + 3.32·13-s − 3.33·23-s − 6/5·25-s − 0.192·27-s + 0.348·33-s − 1.97·37-s − 1.92·39-s − 1.42·49-s − 3.12·59-s − 3.58·61-s + 1.92·69-s + 1.40·73-s + 0.692·75-s + 1/9·81-s + 3.51·83-s − 0.406·97-s − 0.201·99-s + 3.09·107-s − 3.44·109-s + 1.13·111-s + 1.10·117-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(209088\)    =    \(2^{6} \cdot 3^{3} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(13.3316\)
Root analytic conductor: \(1.91082\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 209088,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−8.984065396795677070188454547975, −8.027301198026720221974316484681, −8.017674089261979349140094913788, −7.70654778162849742750783896407, −6.43783880292851993364037339296, −6.41893627610765861044252902088, −5.95505213702500670435098419213, −5.62860857478680063008223727075, −4.77765886027163903937455483278, −4.18020563951979381603420573914, −3.57787548281860178772324547470, −3.34441809775819506943782363931, −1.86785039249087723872477729694, −1.58426641882615255056319824849, 0, 1.58426641882615255056319824849, 1.86785039249087723872477729694, 3.34441809775819506943782363931, 3.57787548281860178772324547470, 4.18020563951979381603420573914, 4.77765886027163903937455483278, 5.62860857478680063008223727075, 5.95505213702500670435098419213, 6.41893627610765861044252902088, 6.43783880292851993364037339296, 7.70654778162849742750783896407, 8.017674089261979349140094913788, 8.027301198026720221974316484681, 8.984065396795677070188454547975

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