Properties

Degree $4$
Conductor $31329$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 2·4-s + 2·6-s − 7·7-s + 3·8-s + 3·9-s + 4·12-s − 8·13-s + 7·14-s + 16-s − 3·17-s − 3·18-s − 5·19-s + 14·21-s − 7·23-s − 6·24-s − 5·25-s + 8·26-s − 4·27-s + 14·28-s + 15·29-s − 31-s − 2·32-s + 3·34-s − 6·36-s − 7·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 4-s + 0.816·6-s − 2.64·7-s + 1.06·8-s + 9-s + 1.15·12-s − 2.21·13-s + 1.87·14-s + 1/4·16-s − 0.727·17-s − 0.707·18-s − 1.14·19-s + 3.05·21-s − 1.45·23-s − 1.22·24-s − 25-s + 1.56·26-s − 0.769·27-s + 2.64·28-s + 2.78·29-s − 0.179·31-s − 0.353·32-s + 0.514·34-s − 36-s − 1.15·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(31329\)    =    \(3^{2} \cdot 59^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{177} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 31329,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
59$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + p T + 25 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 5 T + 13 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 7 T + 57 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
31$C_4$ \( 1 + T - 39 T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 7 T + 75 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 5 T + 57 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 15 T + 139 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 13 T + 153 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 105 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 5 T + 141 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + T + 165 T^{2} + p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.25610894211952283070726136928, −12.22217381661985888528078665499, −11.65351909314581885259103508977, −10.52442417640008856256494621834, −10.22955949647475949358957612481, −10.02773240521845233365231295530, −9.414588112306327463522950750088, −9.371416471870876903956331230307, −8.390133296020820638503016036415, −7.966912615309547791588210516276, −6.87996061281009109379973992105, −6.78262173914783114446061021898, −6.20987270420816684246939244881, −5.54848578879047455243056290371, −4.57452023650093189296413504016, −4.42357676262472025458636139018, −3.36750047154044011266289456885, −2.38059346977599866708154952726, 0, 0, 2.38059346977599866708154952726, 3.36750047154044011266289456885, 4.42357676262472025458636139018, 4.57452023650093189296413504016, 5.54848578879047455243056290371, 6.20987270420816684246939244881, 6.78262173914783114446061021898, 6.87996061281009109379973992105, 7.966912615309547791588210516276, 8.390133296020820638503016036415, 9.371416471870876903956331230307, 9.414588112306327463522950750088, 10.02773240521845233365231295530, 10.22955949647475949358957612481, 10.52442417640008856256494621834, 11.65351909314581885259103508977, 12.22217381661985888528078665499, 12.25610894211952283070726136928

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