Properties

Degree $4$
Conductor $3969$
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 − 3·5-s − 7·7-s − 45·8-s + 9·10-s − 15·11-s − 128·13-s + 21·14-s + 135·16-s + 84·17-s + 16·19-s − 24·20-s + 45·22-s − 84·23-s + 125·25-s + 384·26-s − 56·28-s + 594·29-s + 253·31-s − 360·32-s − 252·34-s + 21·35-s + 316·37-s − 48·38-s + 135·40-s − 720·41-s + ⋯
L(s)  = 1  − 1.06·2-s + 4-s − 0.268·5-s − 0.377·7-s − 1.98·8-s + 0.284·10-s − 0.411·11-s − 2.73·13-s + 0.400·14-s + 2.10·16-s + 1.19·17-s + 0.193·19-s − 0.268·20-s + 0.436·22-s − 0.761·23-s + 25-s + 2.89·26-s − 0.377·28-s + 3.80·29-s + 1.46·31-s − 1.98·32-s − 1.27·34-s + 0.101·35-s + 1.40·37-s − 0.204·38-s + 0.533·40-s − 2.74·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{63} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3969,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.611094\)
\(L(\frac12)\) \(\approx\) \(0.611094\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p T + p^{3} T^{2} \)
good2$C_2^2$ \( 1 + 3 T + T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 3 T - 116 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 15 T - 1106 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 64 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 16 T - 6603 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 84 T - 5111 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 297 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 253 T + 34218 T^{2} - 253 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 316 T + 49203 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 360 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 26 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 30 T - 102923 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 363 T - 17108 T^{2} - 363 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 15 T - 205154 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 118 T - 213057 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 370 T - 163863 T^{2} - 370 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 342 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 362 T - 257973 T^{2} + 362 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 467 T - 274950 T^{2} + 467 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 477 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 906 T + 115867 T^{2} - 906 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 503 T + p^{3} T^{2} )^{2} \)
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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

−15.23103664677289981250143323834, −14.34520857597417977910322597573, −13.85626431182820684584894852303, −12.57850558488342570499673910701, −12.41663850427594300802213263994, −11.83379622712066472602735915478, −11.67947089290166356129346565283, −10.16448849365863446054792451355, −10.12939170901032552824418014413, −9.857748596758354895774400295598, −8.893526795560829037851986070013, −8.147107920806730178073587235006, −7.88830335050283914168109251131, −6.71351177861983897343153583355, −6.65971297350629044580055596238, −5.39935078514453368454433006921, −4.70459420574296370063987558869, −2.93789211719097433935532809527, −2.71962746093523372546122913870, −0.62473778199903321829856653931, 0.62473778199903321829856653931, 2.71962746093523372546122913870, 2.93789211719097433935532809527, 4.70459420574296370063987558869, 5.39935078514453368454433006921, 6.65971297350629044580055596238, 6.71351177861983897343153583355, 7.88830335050283914168109251131, 8.147107920806730178073587235006, 8.893526795560829037851986070013, 9.857748596758354895774400295598, 10.12939170901032552824418014413, 10.16448849365863446054792451355, 11.67947089290166356129346565283, 11.83379622712066472602735915478, 12.41663850427594300802213263994, 12.57850558488342570499673910701, 13.85626431182820684584894852303, 14.34520857597417977910322597573, 15.23103664677289981250143323834

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