Properties

Label 4-7098e2-1.1-c1e2-0-14
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s − 2·5-s − 4·6-s − 2·7-s + 4·8-s + 3·9-s − 4·10-s − 4·11-s − 6·12-s − 4·14-s + 4·15-s + 5·16-s + 4·17-s + 6·18-s + 2·19-s − 6·20-s + 4·21-s − 8·22-s − 8·24-s − 4·25-s − 4·27-s − 6·28-s − 6·29-s + 8·30-s + 2·31-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 1.26·10-s − 1.20·11-s − 1.73·12-s − 1.06·14-s + 1.03·15-s + 5/4·16-s + 0.970·17-s + 1.41·18-s + 0.458·19-s − 1.34·20-s + 0.872·21-s − 1.70·22-s − 1.63·24-s − 4/5·25-s − 0.769·27-s − 1.13·28-s − 1.11·29-s + 1.46·30-s + 0.359·31-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: \(2\)
Selberg data: \((4,\ 50381604,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good5$D_{4}$ \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 2 T - 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 80 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 2 T + 84 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 83 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 20 T + 219 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 20 T + 234 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 175 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T + 192 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−7.60491188952005286021430898257, −7.58145509631131023494676008189, −6.78944660457468836743577281520, −6.73199870497150311580813820600, −6.08929084243319816379316147130, −6.01550537557317169249446694808, −5.53776996053441100246173439024, −5.39864750948483400149937065416, −4.99880740223038422986682391697, −4.49083805521704451209780337618, −4.16418489296889987459682572543, −3.98807138854505203231244587961, −3.33888421009879985532914137220, −3.21240106183492532178608480583, −2.52133487055481954470454049275, −2.41179960947626628687517788851, −1.38733439384469632512183222708, −1.17294995297692128885344185238, 0, 0, 1.17294995297692128885344185238, 1.38733439384469632512183222708, 2.41179960947626628687517788851, 2.52133487055481954470454049275, 3.21240106183492532178608480583, 3.33888421009879985532914137220, 3.98807138854505203231244587961, 4.16418489296889987459682572543, 4.49083805521704451209780337618, 4.99880740223038422986682391697, 5.39864750948483400149937065416, 5.53776996053441100246173439024, 6.01550537557317169249446694808, 6.08929084243319816379316147130, 6.73199870497150311580813820600, 6.78944660457468836743577281520, 7.58145509631131023494676008189, 7.60491188952005286021430898257

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