Properties

Label 4-21e4-1.1-c2e2-0-4
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $144.393$
Root an. cond. $3.46646$
Motivic weight $2$
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  − 2·2-s − 5·4-s + 20·8-s + 22·11-s + 5·16-s − 44·22-s − 56·23-s + 23·25-s − 50·29-s − 118·32-s − 116·37-s + 52·43-s − 110·44-s + 112·46-s − 46·50-s − 62·53-s + 100·58-s + 111·64-s − 104·67-s − 128·71-s + 232·74-s + 34·79-s − 104·86-s + 440·88-s + 280·92-s − 115·100-s + 124·106-s + ⋯
L(s)  = 1  − 2-s − 5/4·4-s + 5/2·8-s + 2·11-s + 5/16·16-s − 2·22-s − 2.43·23-s + 0.919·25-s − 1.72·29-s − 3.68·32-s − 3.13·37-s + 1.20·43-s − 5/2·44-s + 2.43·46-s − 0.919·50-s − 1.16·53-s + 1.72·58-s + 1.73·64-s − 1.55·67-s − 1.80·71-s + 3.13·74-s + 0.430·79-s − 1.20·86-s + 5·88-s + 3.04·92-s − 1.14·100-s + 1.16·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(144.393\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4507931655\)
\(L(\frac12)\) \(\approx\) \(0.4507931655\)
\(L(2)\) 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$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
5$C_2^2$ \( 1 - 23 T^{2} + p^{4} T^{4} \)
11$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 290 T^{2} + p^{4} T^{4} \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{4} T^{4} \)
19$C_2^2$ \( 1 - 710 T^{2} + p^{4} T^{4} \)
23$C_2$ \( ( 1 + 28 T + p^{2} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 25 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 839 T^{2} + p^{4} T^{4} \)
37$C_2$ \( ( 1 + 58 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3350 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 1390 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 + 31 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6887 T^{2} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 7250 T^{2} + p^{4} T^{4} \)
67$C_2$ \( ( 1 + 52 T + p^{2} T^{2} )^{2} \)
71$C_2$ \( ( 1 + 64 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 10610 T^{2} + p^{4} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 10895 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 9494 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 10391 T^{2} + p^{4} T^{4} \)
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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

−10.85280229300935185267603059167, −10.42378921133688084119529269865, −10.27937589761807149491095761633, −9.464250853767643929143422190990, −9.311972758558787273975536959208, −9.032802248052332866035318144247, −8.586855110305760800799446829070, −8.043712140877237405075307288687, −7.71917569020245977946208163348, −7.03599987511220942636917542216, −6.69127876400002716799517939535, −5.70906040619644591605787118543, −5.63029827797143237436178884623, −4.66023674674098844197714639843, −4.26144601823530679895157871110, −3.82138307530585473884559048388, −3.35181295582762685461679930357, −1.69762244042391011219510209170, −1.57833066933262737754955787161, −0.36270571936395091833929555732, 0.36270571936395091833929555732, 1.57833066933262737754955787161, 1.69762244042391011219510209170, 3.35181295582762685461679930357, 3.82138307530585473884559048388, 4.26144601823530679895157871110, 4.66023674674098844197714639843, 5.63029827797143237436178884623, 5.70906040619644591605787118543, 6.69127876400002716799517939535, 7.03599987511220942636917542216, 7.71917569020245977946208163348, 8.043712140877237405075307288687, 8.586855110305760800799446829070, 9.032802248052332866035318144247, 9.311972758558787273975536959208, 9.464250853767643929143422190990, 10.27937589761807149491095761633, 10.42378921133688084119529269865, 10.85280229300935185267603059167

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