Properties

Degree $4$
Conductor $8100$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·4-s − 110·5-s + 296·11-s + 256·16-s − 4.44e3·19-s + 1.76e3·20-s + 8.97e3·25-s − 540·29-s − 4.09e3·31-s + 4.79e3·41-s − 4.73e3·44-s + 8.65e3·49-s − 3.25e4·55-s − 7.94e4·59-s − 8.45e4·61-s − 4.09e3·64-s + 8.49e3·71-s + 7.10e4·76-s − 7.05e4·79-s − 2.81e4·80-s − 1.70e5·89-s + 4.88e5·95-s − 1.43e5·100-s + 8.59e3·101-s + 7.19e4·109-s + 8.64e3·116-s − 2.56e5·121-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.96·5-s + 0.737·11-s + 1/4·16-s − 2.82·19-s + 0.983·20-s + 2.87·25-s − 0.119·29-s − 0.765·31-s + 0.445·41-s − 0.368·44-s + 0.514·49-s − 1.45·55-s − 2.97·59-s − 2.91·61-s − 1/8·64-s + 0.200·71-s + 1.41·76-s − 1.27·79-s − 0.491·80-s − 2.28·89-s + 5.55·95-s − 1.43·100-s + 0.0838·101-s + 0.580·109-s + 0.0596·116-s − 1.59·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{90} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8100,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.167777\)
\(L(\frac12)\) \(\approx\) \(0.167777\)
\(L(\frac{7}{2})\) 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)$
bad2$C_2$ \( 1 + p^{4} T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + 22 p T + p^{5} T^{2} \)
good7$C_2^2$ \( 1 - 8650 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 - 148 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 274730 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 + 1354590 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 + 2220 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 11320170 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + 270 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2048 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 119573530 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 - 2398 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 288754450 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 344584890 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 827605690 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 + 39740 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 42298 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 1669968610 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 4248 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3239892370 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 + 35280 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 7103795010 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 85210 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 7720618690 T^{2} + p^{10} T^{4} \)
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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

−13.18746933666727150660279706217, −12.79815699147473866154743754861, −12.25527056946384563452647572226, −12.01489114720489222406930790558, −11.15492254074759359287940705201, −10.81925603484668640903743472011, −10.44169349266706109599366942044, −9.233753046113321295957484866732, −8.982643058611419284405275498388, −8.333180019243745791419694862899, −7.889540861626861394864905455711, −7.23065331322161692210391381178, −6.57191011569547316844062799138, −5.90046019645149980008757581893, −4.58969947940424697057555459734, −4.38166434214523144874307368998, −3.76574800022741387248522764430, −2.89230975449591630649322238711, −1.51003421998450268117401362950, −0.17294924110467147987588386132, 0.17294924110467147987588386132, 1.51003421998450268117401362950, 2.89230975449591630649322238711, 3.76574800022741387248522764430, 4.38166434214523144874307368998, 4.58969947940424697057555459734, 5.90046019645149980008757581893, 6.57191011569547316844062799138, 7.23065331322161692210391381178, 7.889540861626861394864905455711, 8.333180019243745791419694862899, 8.982643058611419284405275498388, 9.233753046113321295957484866732, 10.44169349266706109599366942044, 10.81925603484668640903743472011, 11.15492254074759359287940705201, 12.01489114720489222406930790558, 12.25527056946384563452647572226, 12.79815699147473866154743754861, 13.18746933666727150660279706217

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