Properties

Label 4-21e4-1.1-c1e2-0-18
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $12.4002$
Root an. cond. $1.87654$
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  + 2-s + 2·4-s + 5·8-s + 4·11-s + 5·16-s + 4·22-s + 8·23-s + 5·25-s − 4·29-s + 10·32-s + 6·37-s − 24·43-s + 8·44-s + 8·46-s + 5·50-s − 10·53-s − 4·58-s + 17·64-s − 4·67-s − 32·71-s + 6·74-s − 8·79-s − 24·86-s + 20·88-s + 16·92-s + 10·100-s − 10·106-s + ⋯
L(s)  = 1  + 0.707·2-s + 4-s + 1.76·8-s + 1.20·11-s + 5/4·16-s + 0.852·22-s + 1.66·23-s + 25-s − 0.742·29-s + 1.76·32-s + 0.986·37-s − 3.65·43-s + 1.20·44-s + 1.17·46-s + 0.707·50-s − 1.37·53-s − 0.525·58-s + 17/8·64-s − 0.488·67-s − 3.79·71-s + 0.697·74-s − 0.900·79-s − 2.58·86-s + 2.13·88-s + 1.66·92-s + 100-s − 0.971·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(12.4002\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.737694151\)
\(L(\frac12)\) \(\approx\) \(3.737694151\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{2} \)
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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

−11.22118699307712351797105705482, −11.18803891444180594550764081904, −10.42260184288219637129287499863, −10.30471708388913224954778517263, −9.423907224831202791519044739523, −9.308040957925004218281811246323, −8.394894458611223772036190976834, −8.287968396041543380977247870036, −7.29139196053559166721759899932, −7.24377531406218404291946005590, −6.62665534373549917236332618591, −6.36579893512035772327223491545, −5.58751412495102888234703243537, −5.01970630315614486339063133602, −4.53634707812790858538234661528, −4.13031728033590283530223709279, −3.17051580992645947470498223898, −2.97309757549757489362482940577, −1.72308377574922347240824843108, −1.36244013126385186572929554582, 1.36244013126385186572929554582, 1.72308377574922347240824843108, 2.97309757549757489362482940577, 3.17051580992645947470498223898, 4.13031728033590283530223709279, 4.53634707812790858538234661528, 5.01970630315614486339063133602, 5.58751412495102888234703243537, 6.36579893512035772327223491545, 6.62665534373549917236332618591, 7.24377531406218404291946005590, 7.29139196053559166721759899932, 8.287968396041543380977247870036, 8.394894458611223772036190976834, 9.308040957925004218281811246323, 9.423907224831202791519044739523, 10.30471708388913224954778517263, 10.42260184288219637129287499863, 11.18803891444180594550764081904, 11.22118699307712351797105705482

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