Properties

Label 4-7098e2-1.1-c1e2-0-9
Degree $4$
Conductor $50381604$
Sign $1$
Analytic cond. $3212.37$
Root an. cond. $7.52846$
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  − 2·2-s − 2·3-s + 3·4-s + 4·5-s + 4·6-s + 2·7-s − 4·8-s + 3·9-s − 8·10-s − 11-s − 6·12-s − 4·14-s − 8·15-s + 5·16-s + 10·17-s − 6·18-s + 5·19-s + 12·20-s − 4·21-s + 2·22-s − 5·23-s + 8·24-s + 2·25-s − 4·27-s + 6·28-s + 5·29-s + 16·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.78·5-s + 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s − 2.52·10-s − 0.301·11-s − 1.73·12-s − 1.06·14-s − 2.06·15-s + 5/4·16-s + 2.42·17-s − 1.41·18-s + 1.14·19-s + 2.68·20-s − 0.872·21-s + 0.426·22-s − 1.04·23-s + 1.63·24-s + 2/5·25-s − 0.769·27-s + 1.13·28-s + 0.928·29-s + 2.92·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50381604\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3212.37\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7098} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50381604,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.458637511\)
\(L(\frac12)\) \(\approx\) \(2.458637511\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
13 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 5 T + 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 5 T + 20 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 13 T + 128 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 19 T + 192 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 7 T + 122 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 15 T + 182 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + T + 134 T^{2} + p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−7.982980989625161124238663572654, −7.83676740257569031736813294144, −7.34326918808562866214833896212, −7.33133297209796264462224475836, −6.60587739163547947906771077375, −6.40921079103539980944260610478, −5.92702329900097779158484049030, −5.66291659458266976546339592980, −5.48475911813519175659179726696, −5.34766821461580158818046158240, −4.72248602363370801909135244286, −4.22643559376834830035243990149, −3.71720204573071596317679520498, −3.17700542512900031679471497779, −2.68052053130926861975025693412, −2.27941211470107522943892843184, −1.68571350499056331620649629374, −1.55014825405553379911654918195, −0.78819259466702459849253676515, −0.74933231328792543555521525060, 0.74933231328792543555521525060, 0.78819259466702459849253676515, 1.55014825405553379911654918195, 1.68571350499056331620649629374, 2.27941211470107522943892843184, 2.68052053130926861975025693412, 3.17700542512900031679471497779, 3.71720204573071596317679520498, 4.22643559376834830035243990149, 4.72248602363370801909135244286, 5.34766821461580158818046158240, 5.48475911813519175659179726696, 5.66291659458266976546339592980, 5.92702329900097779158484049030, 6.40921079103539980944260610478, 6.60587739163547947906771077375, 7.33133297209796264462224475836, 7.34326918808562866214833896212, 7.83676740257569031736813294144, 7.982980989625161124238663572654

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