Properties

Label 4-7098e2-1.1-c1e2-0-12
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 3·4-s + 5-s + 4·6-s − 2·7-s − 4·8-s + 3·9-s − 2·10-s + 11-s − 6·12-s + 4·14-s − 2·15-s + 5·16-s − 8·17-s − 6·18-s + 19-s + 3·20-s + 4·21-s − 2·22-s + 6·23-s + 8·24-s − 5·25-s − 4·27-s − 6·28-s − 4·29-s + 4·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.447·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 0.632·10-s + 0.301·11-s − 1.73·12-s + 1.06·14-s − 0.516·15-s + 5/4·16-s − 1.94·17-s − 1.41·18-s + 0.229·19-s + 0.670·20-s + 0.872·21-s − 0.426·22-s + 1.25·23-s + 1.63·24-s − 25-s − 0.769·27-s − 1.13·28-s − 0.742·29-s + 0.730·30-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: Trivial
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$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 9 T + 90 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 13 T + 98 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 9 T + 162 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 9 T + 174 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 2 T + 150 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 11 T + 204 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 10 T + 202 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−7.50796271085057129483160699590, −7.44361657808607727244718244351, −7.10168982284252194330103154128, −6.74081340930618384109669133126, −6.39780142324244468680838256697, −6.17435354337191395249665941160, −5.76828460978255720077752493668, −5.59096816164183695024004031050, −4.94622864617803131865073911221, −4.68232325932013256492564829049, −4.04388312798977435995292071836, −3.90270579949608689327203002927, −3.22701499135129581390321557480, −2.73116703306235969653789987380, −2.23352502003531102056837657603, −2.04524869212731725829883965286, −1.18249847667525229847670280195, −1.02551404515075722032096849171, 0, 0, 1.02551404515075722032096849171, 1.18249847667525229847670280195, 2.04524869212731725829883965286, 2.23352502003531102056837657603, 2.73116703306235969653789987380, 3.22701499135129581390321557480, 3.90270579949608689327203002927, 4.04388312798977435995292071836, 4.68232325932013256492564829049, 4.94622864617803131865073911221, 5.59096816164183695024004031050, 5.76828460978255720077752493668, 6.17435354337191395249665941160, 6.39780142324244468680838256697, 6.74081340930618384109669133126, 7.10168982284252194330103154128, 7.44361657808607727244718244351, 7.50796271085057129483160699590

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