Properties

Label 4-2850e2-1.1-c1e2-0-0
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s − 2·11-s + 16-s − 2·19-s + 2·29-s + 2·31-s + 36-s − 20·41-s + 2·44-s − 2·49-s − 8·59-s + 10·61-s − 64-s − 12·71-s + 2·76-s + 10·79-s + 81-s − 6·89-s + 2·99-s − 24·101-s − 12·109-s − 2·116-s − 19·121-s − 2·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 0.603·11-s + 1/4·16-s − 0.458·19-s + 0.371·29-s + 0.359·31-s + 1/6·36-s − 3.12·41-s + 0.301·44-s − 2/7·49-s − 1.04·59-s + 1.28·61-s − 1/8·64-s − 1.42·71-s + 0.229·76-s + 1.12·79-s + 1/9·81-s − 0.635·89-s + 0.201·99-s − 2.38·101-s − 1.14·109-s − 0.185·116-s − 1.72·121-s − 0.179·124-s + 0.0887·127-s + 0.0873·131-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

\(L(1)\) \(\approx\) \(0.5002654942\)
\(L(\frac12)\) \(\approx\) \(0.5002654942\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−9.019547245185164822498182028485, −8.565921616009679837369866078455, −8.131249668215418664036003275647, −8.050809400799723817092333708446, −7.58917468173779138562205069234, −6.99714789244424905392739362598, −6.60348240883292456273867449923, −6.49482679930193600594389305397, −5.83854768007935758634163650597, −5.34297716034151522974190983796, −5.20778686681708888727461539453, −4.72034007583467629561633064723, −4.30351690166654931702167538274, −3.75229426427201101216240727280, −3.43047708817711198218963468974, −2.78031562473522690595690063204, −2.53489072780661097603399962009, −1.71613523417364555227614327199, −1.27978972478432578826574830260, −0.23279685588340032062251358667, 0.23279685588340032062251358667, 1.27978972478432578826574830260, 1.71613523417364555227614327199, 2.53489072780661097603399962009, 2.78031562473522690595690063204, 3.43047708817711198218963468974, 3.75229426427201101216240727280, 4.30351690166654931702167538274, 4.72034007583467629561633064723, 5.20778686681708888727461539453, 5.34297716034151522974190983796, 5.83854768007935758634163650597, 6.49482679930193600594389305397, 6.60348240883292456273867449923, 6.99714789244424905392739362598, 7.58917468173779138562205069234, 8.050809400799723817092333708446, 8.131249668215418664036003275647, 8.565921616009679837369866078455, 9.019547245185164822498182028485

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