Properties

Degree $4$
Conductor $21622500$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s − 8·11-s + 16-s − 8·19-s − 12·29-s − 2·31-s + 36-s + 20·41-s + 8·44-s + 14·49-s + 24·59-s − 4·61-s − 64-s + 8·76-s + 81-s + 28·89-s + 8·99-s + 12·101-s + 36·109-s + 12·116-s + 26·121-s + 2·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 2.41·11-s + 1/4·16-s − 1.83·19-s − 2.22·29-s − 0.359·31-s + 1/6·36-s + 3.12·41-s + 1.20·44-s + 2·49-s + 3.12·59-s − 0.512·61-s − 1/8·64-s + 0.917·76-s + 1/9·81-s + 2.96·89-s + 0.804·99-s + 1.19·101-s + 3.44·109-s + 1.11·116-s + 2.36·121-s + 0.179·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21622500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 31^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{4650} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21622500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.238427401\)
\(L(\frac12)\) \(\approx\) \(1.238427401\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
31$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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

−8.525647481673992696530012265165, −8.131172091433443907611229110211, −7.64225774108516106770381217401, −7.52881854441892907560908322998, −7.28985638622406679814529895354, −6.73066408545719865337403776170, −6.03265036783062671656038008801, −5.84179288751046260251331246618, −5.73308360394574246755242445904, −5.05503011365676459693430618962, −4.97036767889182386816932168076, −4.36700328846565953683990333043, −3.93407613333581363368019575718, −3.72116992769178786307264091475, −3.06084798008186798510762988705, −2.51835469249194975979516482251, −2.10672157202498530092111965838, −2.10071159374473486521558737460, −0.75200811348567704805352024882, −0.42491837607475486422221526359, 0.42491837607475486422221526359, 0.75200811348567704805352024882, 2.10071159374473486521558737460, 2.10672157202498530092111965838, 2.51835469249194975979516482251, 3.06084798008186798510762988705, 3.72116992769178786307264091475, 3.93407613333581363368019575718, 4.36700328846565953683990333043, 4.97036767889182386816932168076, 5.05503011365676459693430618962, 5.73308360394574246755242445904, 5.84179288751046260251331246618, 6.03265036783062671656038008801, 6.73066408545719865337403776170, 7.28985638622406679814529895354, 7.52881854441892907560908322998, 7.64225774108516106770381217401, 8.131172091433443907611229110211, 8.525647481673992696530012265165

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