Properties

Label 4-2850e2-1.1-c1e2-0-7
Degree $4$
Conductor $8122500$
Sign $1$
Analytic cond. $517.897$
Root an. cond. $4.77046$
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  − 4-s − 9-s + 6·11-s + 16-s − 2·19-s − 6·29-s − 14·31-s + 36-s − 12·41-s − 6·44-s + 10·49-s − 12·59-s − 2·61-s − 64-s + 2·76-s + 26·79-s + 81-s + 30·89-s − 6·99-s + 12·101-s − 4·109-s + 6·116-s + 5·121-s + 14·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 1.80·11-s + 1/4·16-s − 0.458·19-s − 1.11·29-s − 2.51·31-s + 1/6·36-s − 1.87·41-s − 0.904·44-s + 10/7·49-s − 1.56·59-s − 0.256·61-s − 1/8·64-s + 0.229·76-s + 2.92·79-s + 1/9·81-s + 3.17·89-s − 0.603·99-s + 1.19·101-s − 0.383·109-s + 0.557·116-s + 5/11·121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8122500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(517.897\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.434076097\)
\(L(\frac12)\) \(\approx\) \(1.434076097\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 59 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.960817171424932230850591668881, −8.925817151104560719677690922187, −8.203689491785621569435049598151, −7.85867432769430594325865061508, −7.49794897295285178928755009453, −7.06360070851082435006210612849, −6.69293565861219500582445708262, −6.28568428421864848302951154897, −5.97489717608363614423259902517, −5.51763079022218854793002672080, −4.95256193733036729163702073838, −4.88125390593818221945071210739, −3.93595653881679154302446511767, −3.91615768826487210993454157480, −3.52307290822718916342623165467, −3.06242873533281056358605333291, −2.05829890358335335103380825973, −1.92977440294093514106609649805, −1.24117565967959199251996470866, −0.40478713874505623865881763221, 0.40478713874505623865881763221, 1.24117565967959199251996470866, 1.92977440294093514106609649805, 2.05829890358335335103380825973, 3.06242873533281056358605333291, 3.52307290822718916342623165467, 3.91615768826487210993454157480, 3.93595653881679154302446511767, 4.88125390593818221945071210739, 4.95256193733036729163702073838, 5.51763079022218854793002672080, 5.97489717608363614423259902517, 6.28568428421864848302951154897, 6.69293565861219500582445708262, 7.06360070851082435006210612849, 7.49794897295285178928755009453, 7.85867432769430594325865061508, 8.203689491785621569435049598151, 8.925817151104560719677690922187, 8.960817171424932230850591668881

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