Properties

Label 4-21e4-1.1-c1e2-0-21
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s + 2·4-s − 5-s + 3·6-s − 5·8-s + 6·9-s + 10-s − 5·11-s − 6·12-s − 5·13-s + 3·15-s + 5·16-s − 6·17-s − 6·18-s − 2·19-s − 2·20-s + 5·22-s − 3·23-s + 15·24-s + 5·25-s + 5·26-s − 9·27-s + 29-s − 3·30-s − 10·32-s + 15·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 4-s − 0.447·5-s + 1.22·6-s − 1.76·8-s + 2·9-s + 0.316·10-s − 1.50·11-s − 1.73·12-s − 1.38·13-s + 0.774·15-s + 5/4·16-s − 1.45·17-s − 1.41·18-s − 0.458·19-s − 0.447·20-s + 1.06·22-s − 0.625·23-s + 3.06·24-s + 25-s + 0.980·26-s − 1.73·27-s + 0.185·29-s − 0.547·30-s − 1.76·32-s + 2.61·33-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: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 194481,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
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

−10.79986000589164937841870418537, −10.70371277507289663796721055705, −10.04680456843492734903442569177, −9.840309209941096572813486996248, −9.015134875689948125772806480985, −8.812356764534351378782297816606, −7.972592848908292992146851702832, −7.57778244473326964945080354131, −7.20582544937681542002073887943, −6.59997082647348295533167961426, −6.24708362972419236706789795104, −5.87729487185656073521172061780, −5.13918503241466453936853480635, −4.74318850001069861386297510355, −4.27111317857515609001967569529, −2.98802852032987997401209236625, −2.65308002061121635953759547720, −1.71499801023174085083386906158, 0, 0, 1.71499801023174085083386906158, 2.65308002061121635953759547720, 2.98802852032987997401209236625, 4.27111317857515609001967569529, 4.74318850001069861386297510355, 5.13918503241466453936853480635, 5.87729487185656073521172061780, 6.24708362972419236706789795104, 6.59997082647348295533167961426, 7.20582544937681542002073887943, 7.57778244473326964945080354131, 7.972592848908292992146851702832, 8.812356764534351378782297816606, 9.015134875689948125772806480985, 9.840309209941096572813486996248, 10.04680456843492734903442569177, 10.70371277507289663796721055705, 10.79986000589164937841870418537

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