Properties

Label 4-21e4-1.1-c1e2-0-19
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s + 2·4-s + 2·5-s − 3·6-s − 5·8-s + 6·9-s − 2·10-s + 10·11-s + 6·12-s − 5·13-s + 6·15-s + 5·16-s + 3·17-s − 6·18-s + 19-s + 4·20-s − 10·22-s + 6·23-s − 15·24-s − 7·25-s + 5·26-s + 9·27-s + 29-s − 6·30-s − 10·32-s + 30·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 4-s + 0.894·5-s − 1.22·6-s − 1.76·8-s + 2·9-s − 0.632·10-s + 3.01·11-s + 1.73·12-s − 1.38·13-s + 1.54·15-s + 5/4·16-s + 0.727·17-s − 1.41·18-s + 0.229·19-s + 0.894·20-s − 2.13·22-s + 1.25·23-s − 3.06·24-s − 7/5·25-s + 0.980·26-s + 1.73·27-s + 0.185·29-s − 1.09·30-s − 1.76·32-s + 5.22·33-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.241226056\)
\(L(\frac12)\) \(\approx\) \(3.241226056\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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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

−11.67270127515268801840036541869, −10.68464167078627835701625326494, −10.05889829499358720750331639581, −9.813220415498522251491742169714, −9.385078647897478007497369838284, −9.207429231171912215799198859884, −8.666517176668105822070017843953, −8.584233361275081863493018719485, −7.47292159610009580697029651338, −7.36975048322615706487256264248, −6.91988566287984603458181768886, −6.26463411179910811009541442903, −6.01572280651773631726681495099, −5.20137328974966819605129078249, −4.22808360698426185127864388297, −3.75520227418099950711137063037, −3.05696271788775605588001218213, −2.69016880952267537302195258300, −1.74641034693958679014655227369, −1.45886971208607556344196644717, 1.45886971208607556344196644717, 1.74641034693958679014655227369, 2.69016880952267537302195258300, 3.05696271788775605588001218213, 3.75520227418099950711137063037, 4.22808360698426185127864388297, 5.20137328974966819605129078249, 6.01572280651773631726681495099, 6.26463411179910811009541442903, 6.91988566287984603458181768886, 7.36975048322615706487256264248, 7.47292159610009580697029651338, 8.584233361275081863493018719485, 8.666517176668105822070017843953, 9.207429231171912215799198859884, 9.385078647897478007497369838284, 9.813220415498522251491742169714, 10.05889829499358720750331639581, 10.68464167078627835701625326494, 11.67270127515268801840036541869

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