Properties

Degree $4$
Conductor $194481$
Sign $1$
Motivic weight $1$
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 + 2·4-s + 2·5-s − 5·8-s − 2·10-s + 4·11-s + 4·13-s + 5·16-s + 6·17-s + 4·19-s + 4·20-s − 4·22-s + 5·25-s − 4·26-s + 4·29-s − 10·32-s − 6·34-s − 6·37-s − 4·38-s − 10·40-s + 4·41-s − 8·43-s + 8·44-s − 5·50-s + 8·52-s + 6·53-s + 8·55-s + ⋯
L(s)  = 1  − 0.707·2-s + 4-s + 0.894·5-s − 1.76·8-s − 0.632·10-s + 1.20·11-s + 1.10·13-s + 5/4·16-s + 1.45·17-s + 0.917·19-s + 0.894·20-s − 0.852·22-s + 25-s − 0.784·26-s + 0.742·29-s − 1.76·32-s − 1.02·34-s − 0.986·37-s − 0.648·38-s − 1.58·40-s + 0.624·41-s − 1.21·43-s + 1.20·44-s − 0.707·50-s + 1.10·52-s + 0.824·53-s + 1.07·55-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: $1$
Motivic weight: \(1\)
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

\(L(1)\) \(\approx\) \(1.86061\)
\(L(\frac12)\) \(\approx\) \(1.86061\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
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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

−11.42371735861110295501405281529, −10.89860858961831135845621597150, −10.17990254695275946028945551699, −10.12863263573002084837923822478, −9.402543631248413595190775792406, −9.245938166960791646531176792884, −8.604500326453477787828110058279, −8.448713088067621153396401594024, −7.66969197391076211224592416394, −7.09363382810483094350572534532, −6.71877183662010098343453382951, −6.18650228057432821400645670841, −5.84256514333728356367828773146, −5.45336470547072959109809118021, −4.63372453164820815437507166258, −3.55722881914239991028849963167, −3.31773101016116623867932707704, −2.63453838237042863909515533895, −1.60812344594404046507223451442, −1.11564626932803369620485645805, 1.11564626932803369620485645805, 1.60812344594404046507223451442, 2.63453838237042863909515533895, 3.31773101016116623867932707704, 3.55722881914239991028849963167, 4.63372453164820815437507166258, 5.45336470547072959109809118021, 5.84256514333728356367828773146, 6.18650228057432821400645670841, 6.71877183662010098343453382951, 7.09363382810483094350572534532, 7.66969197391076211224592416394, 8.448713088067621153396401594024, 8.604500326453477787828110058279, 9.245938166960791646531176792884, 9.402543631248413595190775792406, 10.12863263573002084837923822478, 10.17990254695275946028945551699, 10.89860858961831135845621597150, 11.42371735861110295501405281529

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