Properties

Label 4-1850e2-1.1-c1e2-0-9
Degree $4$
Conductor $3422500$
Sign $1$
Analytic cond. $218.221$
Root an. cond. $3.84347$
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 + 2·9-s + 6·11-s + 16-s − 4·19-s − 6·29-s + 10·31-s − 2·36-s + 6·41-s − 6·44-s + 13·49-s − 2·61-s − 64-s + 12·71-s + 4·76-s − 16·79-s − 5·81-s + 12·89-s + 12·99-s − 12·101-s − 22·109-s + 6·116-s + 5·121-s − 10·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s + 2/3·9-s + 1.80·11-s + 1/4·16-s − 0.917·19-s − 1.11·29-s + 1.79·31-s − 1/3·36-s + 0.937·41-s − 0.904·44-s + 13/7·49-s − 0.256·61-s − 1/8·64-s + 1.42·71-s + 0.458·76-s − 1.80·79-s − 5/9·81-s + 1.27·89-s + 1.20·99-s − 1.19·101-s − 2.10·109-s + 0.557·116-s + 5/11·121-s − 0.898·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3422500\)    =    \(2^{2} \cdot 5^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(218.221\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3422500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.492450406\)
\(L(\frac12)\) \(\approx\) \(2.492450406\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
37$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
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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

−9.469795911187362942598550550258, −8.950746532704195704092773024545, −8.899177216290367638252314334019, −8.129848952162743451971820262235, −8.109686740664073178919541779159, −7.42751653773487178162056300692, −6.88678194097256094925194069278, −6.86002888593224999857280811535, −6.15597997467289713825454436392, −6.01244020853163183918690084216, −5.39845382891670283639741923816, −4.90775290295400081722568089157, −4.23207593060098564990426900551, −4.13073344291865515346216601470, −3.89246028242719574943863524280, −3.10192255222434719448342467617, −2.54604595404544582857000315103, −1.84889183125781260900045379826, −1.29159593380292501720843578493, −0.65293521367922629127572194155, 0.65293521367922629127572194155, 1.29159593380292501720843578493, 1.84889183125781260900045379826, 2.54604595404544582857000315103, 3.10192255222434719448342467617, 3.89246028242719574943863524280, 4.13073344291865515346216601470, 4.23207593060098564990426900551, 4.90775290295400081722568089157, 5.39845382891670283639741923816, 6.01244020853163183918690084216, 6.15597997467289713825454436392, 6.86002888593224999857280811535, 6.88678194097256094925194069278, 7.42751653773487178162056300692, 8.109686740664073178919541779159, 8.129848952162743451971820262235, 8.899177216290367638252314334019, 8.950746532704195704092773024545, 9.469795911187362942598550550258

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