Properties

Label 4-21e4-1.1-c2e2-0-12
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 5·4-s + 6·5-s + 3·11-s + 15·12-s − 4·13-s + 18·15-s + 9·16-s − 27·17-s + 11·19-s + 30·20-s + 48·23-s − 25-s − 27·27-s + 78·29-s − 64·31-s + 9·33-s + 34·37-s − 12·39-s + 21·41-s + 61·43-s + 15·44-s + 27·48-s − 81·51-s − 20·52-s + 18·55-s + 33·57-s + ⋯
L(s)  = 1  + 3-s + 5/4·4-s + 6/5·5-s + 3/11·11-s + 5/4·12-s − 0.307·13-s + 6/5·15-s + 9/16·16-s − 1.58·17-s + 0.578·19-s + 3/2·20-s + 2.08·23-s − 0.0399·25-s − 27-s + 2.68·29-s − 2.06·31-s + 3/11·33-s + 0.918·37-s − 0.307·39-s + 0.512·41-s + 1.41·43-s + 0.340·44-s + 9/16·48-s − 1.58·51-s − 0.384·52-s + 0.327·55-s + 0.578·57-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

\(L(\frac{3}{2})\) \(\approx\) \(6.070636319\)
\(L(\frac12)\) \(\approx\) \(6.070636319\)
\(L(2)\) 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$C_2$ \( 1 - p T + p^{2} T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 - 5 T^{2} + p^{4} T^{4} \)
5$C_2^2$ \( 1 - 6 T + 37 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2^2$ \( 1 - 3 T + 124 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \)
13$C_2^2$ \( 1 + 4 T - 153 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2^2$ \( 1 + 27 T + 532 T^{2} + 27 p^{2} T^{3} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 37 T + p^{2} T^{2} )( 1 + 26 T + p^{2} T^{2} ) \)
23$C_2^2$ \( 1 - 48 T + 1297 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 78 T + 2869 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 32 T + p^{2} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 34 T - 213 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 21 T + 1828 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 83 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \)
47$C_2^2$ \( 1 - 2066 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
59$C_2^2$ \( 1 - 4439 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 56 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 31 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 9110 T^{2} + p^{4} T^{4} \)
73$C_2^2$ \( 1 - 65 T - 1104 T^{2} - 65 p^{2} T^{3} + p^{4} T^{4} \)
79$C_2$ \( ( 1 - 38 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 84 T + 9241 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 216 T + 23473 T^{2} - 216 p^{2} T^{3} + p^{4} T^{4} \)
97$C_2^2$ \( 1 + 115 T + 3816 T^{2} + 115 p^{2} T^{3} + p^{4} T^{4} \)
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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

−11.00754095712390452394510976363, −10.72626311565197136311345174753, −10.39333405544781594846287619152, −9.472833767596023370678049380539, −9.350210388544497635464041846027, −9.024614596675446818489411589690, −8.582096385291823553051281871797, −7.72040367457546917246379825629, −7.57731871236040721163193300126, −6.78805337109166483350857607342, −6.66190885916292884001096935243, −6.00302568854295064154686689533, −5.62543646333901884816140897943, −4.79733193995409475503293225378, −4.38249184500569113864135532866, −3.25578981045021719230982005571, −2.97317506794262574674052659010, −2.18202305963022023835693674072, −2.09199981937909895856030485874, −0.985588327459375076197669006356, 0.985588327459375076197669006356, 2.09199981937909895856030485874, 2.18202305963022023835693674072, 2.97317506794262574674052659010, 3.25578981045021719230982005571, 4.38249184500569113864135532866, 4.79733193995409475503293225378, 5.62543646333901884816140897943, 6.00302568854295064154686689533, 6.66190885916292884001096935243, 6.78805337109166483350857607342, 7.57731871236040721163193300126, 7.72040367457546917246379825629, 8.582096385291823553051281871797, 9.024614596675446818489411589690, 9.350210388544497635464041846027, 9.472833767596023370678049380539, 10.39333405544781594846287619152, 10.72626311565197136311345174753, 11.00754095712390452394510976363

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