Properties

Label 4-299475-1.1-c1e2-0-14
Degree $4$
Conductor $299475$
Sign $-1$
Analytic cond. $19.0947$
Root an. cond. $2.09039$
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 + 5-s − 4·7-s − 2·9-s + 11-s + 8·13-s − 15-s − 4·16-s + 4·21-s − 2·23-s − 4·25-s + 5·27-s + 14·31-s − 33-s − 4·35-s − 8·39-s − 16·41-s − 12·43-s − 2·45-s + 16·47-s + 4·48-s − 2·49-s − 12·53-s + 55-s + 8·63-s + 8·65-s + 2·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.51·7-s − 2/3·9-s + 0.301·11-s + 2.21·13-s − 0.258·15-s − 16-s + 0.872·21-s − 0.417·23-s − 4/5·25-s + 0.962·27-s + 2.51·31-s − 0.174·33-s − 0.676·35-s − 1.28·39-s − 2.49·41-s − 1.82·43-s − 0.298·45-s + 2.33·47-s + 0.577·48-s − 2/7·49-s − 1.64·53-s + 0.134·55-s + 1.00·63-s + 0.992·65-s + 0.240·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 299475 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299475 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(299475\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(19.0947\)
Root analytic conductor: \(2.09039\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{299475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 299475,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_2$ \( 1 + T + p T^{2} \)
5$C_2$ \( 1 - T + p T^{2} \)
11$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−8.603539619290756001226038948684, −8.345209607147577142729102878076, −7.74496311196663047954564318032, −6.78531143296656324345556882625, −6.45204384569677222710524497943, −6.36261389471308870138602900888, −5.98440833216241163281544616906, −5.29166627611063586116446302527, −4.74017388832099283147560878352, −4.01222477715404816999997837461, −3.39492990778572707531432544664, −3.08408694735404908432700286060, −2.16368511729015593620864857203, −1.21009210304083516826165257888, 0, 1.21009210304083516826165257888, 2.16368511729015593620864857203, 3.08408694735404908432700286060, 3.39492990778572707531432544664, 4.01222477715404816999997837461, 4.74017388832099283147560878352, 5.29166627611063586116446302527, 5.98440833216241163281544616906, 6.36261389471308870138602900888, 6.45204384569677222710524497943, 6.78531143296656324345556882625, 7.74496311196663047954564318032, 8.345209607147577142729102878076, 8.603539619290756001226038948684

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