Properties

Label 4-2850e2-1.1-c1e2-0-27
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 2·7-s − 4·8-s + 3·9-s + 6·12-s − 6·13-s + 4·14-s + 5·16-s + 2·17-s − 6·18-s + 2·19-s − 4·21-s − 10·23-s − 8·24-s + 12·26-s + 4·27-s − 6·28-s − 8·29-s + 2·31-s − 6·32-s − 4·34-s + 9·36-s + 4·37-s − 4·38-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 1.73·12-s − 1.66·13-s + 1.06·14-s + 5/4·16-s + 0.485·17-s − 1.41·18-s + 0.458·19-s − 0.872·21-s − 2.08·23-s − 1.63·24-s + 2.35·26-s + 0.769·27-s − 1.13·28-s − 1.48·29-s + 0.359·31-s − 1.06·32-s − 0.685·34-s + 3/2·36-s + 0.657·37-s − 0.648·38-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: \(2\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 20 T + 219 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 107 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 10 T + 140 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 22 T + 255 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 111 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 16 T + 155 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 139 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 10 T + 72 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
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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

−8.536488400654160247068817134486, −8.192172523176057870198739897306, −7.982956563127859689007609208854, −7.57734907748221602823051809229, −7.14246015732094541955822276108, −7.13595189233618878586300889518, −6.30323596442009799741462372823, −6.28975982321683365853817686689, −5.55113061853396545761001209521, −5.24626868832154175838195164346, −4.55930675844389283910609832028, −4.11592309554262386120279484126, −3.54332422210795593288400057851, −3.22244691563603818976206701499, −2.56942227413473388111562399081, −2.49417601257485511637456637290, −1.56375711190024545225674481001, −1.53458518684767142357198290241, 0, 0, 1.53458518684767142357198290241, 1.56375711190024545225674481001, 2.49417601257485511637456637290, 2.56942227413473388111562399081, 3.22244691563603818976206701499, 3.54332422210795593288400057851, 4.11592309554262386120279484126, 4.55930675844389283910609832028, 5.24626868832154175838195164346, 5.55113061853396545761001209521, 6.28975982321683365853817686689, 6.30323596442009799741462372823, 7.13595189233618878586300889518, 7.14246015732094541955822276108, 7.57734907748221602823051809229, 7.982956563127859689007609208854, 8.192172523176057870198739897306, 8.536488400654160247068817134486

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