Properties

Label 4-4400e2-1.1-c1e2-0-19
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·9-s + 2·11-s + 16·19-s − 10·31-s + 10·49-s + 6·59-s − 8·61-s − 30·71-s + 4·79-s + 16·81-s + 18·89-s + 10·99-s + 36·101-s − 4·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 80·171-s + 173-s + ⋯
L(s)  = 1  + 5/3·9-s + 0.603·11-s + 3.67·19-s − 1.79·31-s + 10/7·49-s + 0.781·59-s − 1.02·61-s − 3.56·71-s + 0.450·79-s + 16/9·81-s + 1.90·89-s + 1.00·99-s + 3.58·101-s − 0.383·109-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 + 0.769·169-s + 6.11·171-s + 0.0760·173-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

\(L(1)\) \(\approx\) \(4.661680666\)
\(L(\frac12)\) \(\approx\) \(4.661680666\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 133 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
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

−8.695463446717145009632970760488, −7.997827130729703101699967579319, −7.50700605595504378349956440621, −7.49509023004236385675479701520, −7.21483224794068178871225338496, −6.97707025066699430532783075772, −6.34039776536889302371137452106, −5.95582056229150782637958848524, −5.47587460657565924731547680587, −5.36778806196894670622615506095, −4.65701264648135938152007293543, −4.56528802534845176075101829937, −3.96312521004927619865570080613, −3.42208638280408150030517365734, −3.38934568960171691729558107417, −2.80695400218705983522056045701, −1.97245136586625596906868444876, −1.66434073440817626084174201772, −1.09312249414062706774388644984, −0.71487897591103813824125481204, 0.71487897591103813824125481204, 1.09312249414062706774388644984, 1.66434073440817626084174201772, 1.97245136586625596906868444876, 2.80695400218705983522056045701, 3.38934568960171691729558107417, 3.42208638280408150030517365734, 3.96312521004927619865570080613, 4.56528802534845176075101829937, 4.65701264648135938152007293543, 5.36778806196894670622615506095, 5.47587460657565924731547680587, 5.95582056229150782637958848524, 6.34039776536889302371137452106, 6.97707025066699430532783075772, 7.21483224794068178871225338496, 7.49509023004236385675479701520, 7.50700605595504378349956440621, 7.997827130729703101699967579319, 8.695463446717145009632970760488

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