Properties

Degree $4$
Conductor $194481$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 8·4-s − 18·5-s − 45·8-s + 54·10-s − 36·11-s − 68·13-s + 135·16-s + 42·17-s + 124·19-s − 144·20-s + 108·22-s + 125·25-s + 204·26-s − 204·29-s + 160·31-s − 360·32-s − 126·34-s − 398·37-s − 372·38-s + 810·40-s + 636·41-s − 536·43-s − 288·44-s + 240·47-s − 375·50-s − 544·52-s + ⋯
L(s)  = 1  − 1.06·2-s + 4-s − 1.60·5-s − 1.98·8-s + 1.70·10-s − 0.986·11-s − 1.45·13-s + 2.10·16-s + 0.599·17-s + 1.49·19-s − 1.60·20-s + 1.04·22-s + 25-s + 1.53·26-s − 1.30·29-s + 0.926·31-s − 1.98·32-s − 0.635·34-s − 1.76·37-s − 1.58·38-s + 3.20·40-s + 2.42·41-s − 1.90·43-s − 0.986·44-s + 0.744·47-s − 1.06·50-s − 1.45·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

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

Particular Values

\(L(2)\) \(\approx\) \(0.1310532850\)
\(L(\frac12)\) \(\approx\) \(0.1310532850\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 + 3 T + T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 18 T + 199 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 36 T - 35 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 34 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 42 T - 3149 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 124 T + 8517 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 102 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 160 T - 4191 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 398 T + 107751 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 318 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 268 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 240 T - 46223 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 498 T + 99127 T^{2} + 498 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 132 T - 187955 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 398 T - 68577 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 92 T - 292299 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 720 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 502 T - 137013 T^{2} - 502 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 1024 T + 555537 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 204 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 354 T - 579653 T^{2} - 354 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 286 T + p^{3} T^{2} )^{2} \)
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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

−10.72244269458304830184612150227, −10.69725525511903010211118579835, −9.697469467794690324601294146015, −9.652805353293206174294071375458, −9.295288168287308177185461450481, −8.532309098398154091855574850792, −8.080118917161985768171813237884, −7.78902893496095748586125694779, −7.44750598350123986931806370584, −7.06680650051527840579043149330, −6.41835466325026410003292129222, −5.73381812900763528845855673376, −5.17805637587241581848038960416, −4.82719764011178492039448634830, −3.66704798221737614130064441882, −3.44563953535668772175388960122, −2.78478356469861248565595900626, −2.18558658586899059421386427078, −0.971526907687978120401894388021, −0.17148708271327106614573697382, 0.17148708271327106614573697382, 0.971526907687978120401894388021, 2.18558658586899059421386427078, 2.78478356469861248565595900626, 3.44563953535668772175388960122, 3.66704798221737614130064441882, 4.82719764011178492039448634830, 5.17805637587241581848038960416, 5.73381812900763528845855673376, 6.41835466325026410003292129222, 7.06680650051527840579043149330, 7.44750598350123986931806370584, 7.78902893496095748586125694779, 8.080118917161985768171813237884, 8.532309098398154091855574850792, 9.295288168287308177185461450481, 9.652805353293206174294071375458, 9.697469467794690324601294146015, 10.69725525511903010211118579835, 10.72244269458304830184612150227

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