Properties

Label 4-4400e2-1.1-c1e2-0-16
Degree $4$
Conductor $19360000$
Sign $1$
Analytic cond. $1234.41$
Root an. cond. $5.92740$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 2·11-s + 8·19-s + 16·29-s + 14·31-s + 8·41-s + 10·49-s − 2·59-s + 8·61-s − 6·71-s + 4·79-s − 30·89-s − 6·99-s − 20·101-s + 28·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s − 24·171-s + ⋯
L(s)  = 1  − 9-s + 0.603·11-s + 1.83·19-s + 2.97·29-s + 2.51·31-s + 1.24·41-s + 10/7·49-s − 0.260·59-s + 1.02·61-s − 0.712·71-s + 0.450·79-s − 3.17·89-s − 0.603·99-s − 1.99·101-s + 2.68·109-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s − 1.83·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(19360000\)    =    \(2^{8} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1234.41\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 19360000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.616801486\)
\(L(\frac12)\) \(\approx\) \(3.616801486\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
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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

−8.591158913313704180525562879618, −8.210438531955501446103592766434, −7.951431383324858364901210758269, −7.41209920928680676984447504326, −7.09981035968807620092504832390, −6.70388946021883691305078858621, −6.26457842295921585138265576609, −6.12672976890198425998295788291, −5.47895167395053194509541917472, −5.39381535128777363073752519237, −4.69262992226800341002375474446, −4.53001543068568994012316134246, −4.05569850635683108654609145824, −3.54331587082993759977051842019, −2.91708226982043547549471154008, −2.69487002432059801616202550248, −2.57937755004782498892208386134, −1.48746709163632682711682285924, −1.00102072246229447522872074416, −0.67020504408855193567243288286, 0.67020504408855193567243288286, 1.00102072246229447522872074416, 1.48746709163632682711682285924, 2.57937755004782498892208386134, 2.69487002432059801616202550248, 2.91708226982043547549471154008, 3.54331587082993759977051842019, 4.05569850635683108654609145824, 4.53001543068568994012316134246, 4.69262992226800341002375474446, 5.39381535128777363073752519237, 5.47895167395053194509541917472, 6.12672976890198425998295788291, 6.26457842295921585138265576609, 6.70388946021883691305078858621, 7.09981035968807620092504832390, 7.41209920928680676984447504326, 7.951431383324858364901210758269, 8.210438531955501446103592766434, 8.591158913313704180525562879618

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