Properties

Label 4-418e2-1.1-c1e2-0-5
Degree $4$
Conductor $174724$
Sign $1$
Analytic cond. $11.1405$
Root an. cond. $1.82695$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 3·4-s + 4·5-s − 2·6-s + 3·7-s − 4·8-s − 9-s − 8·10-s + 2·11-s + 3·12-s − 3·13-s − 6·14-s + 4·15-s + 5·16-s + 3·17-s + 2·18-s − 2·19-s + 12·20-s + 3·21-s − 4·22-s + 7·23-s − 4·24-s + 2·25-s + 6·26-s + 9·28-s + 29-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 3/2·4-s + 1.78·5-s − 0.816·6-s + 1.13·7-s − 1.41·8-s − 1/3·9-s − 2.52·10-s + 0.603·11-s + 0.866·12-s − 0.832·13-s − 1.60·14-s + 1.03·15-s + 5/4·16-s + 0.727·17-s + 0.471·18-s − 0.458·19-s + 2.68·20-s + 0.654·21-s − 0.852·22-s + 1.45·23-s − 0.816·24-s + 2/5·25-s + 1.17·26-s + 1.70·28-s + 0.185·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(174724\)    =    \(2^{2} \cdot 11^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(11.1405\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 174724,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.753013993\)
\(L(\frac12)\) \(\approx\) \(1.753013993\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 3 T + 98 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 7 T + 120 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 10 T + 186 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 6 T + 186 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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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

−11.11878951294201265363620957806, −10.86878081460772726630137738357, −10.36963481193907933889176008593, −9.938850994382666261086169202310, −9.510415366877337692324840405230, −9.269548033660763706910973034590, −8.700309598822903808568055572210, −8.497014919566527020118211833463, −7.74131100109873532120571409940, −7.55971591370878823469256548776, −6.79288055299980896065481533675, −6.45450092424796220281613801753, −5.65341339453139235578459003044, −5.50495225903859906873679895599, −4.74359146288740557772427399281, −3.91222257259820018576286398452, −2.72121495121319900109329876429, −2.58434676725530789994076150184, −1.72927659535003427204067869921, −1.18697936166448359819284346779, 1.18697936166448359819284346779, 1.72927659535003427204067869921, 2.58434676725530789994076150184, 2.72121495121319900109329876429, 3.91222257259820018576286398452, 4.74359146288740557772427399281, 5.50495225903859906873679895599, 5.65341339453139235578459003044, 6.45450092424796220281613801753, 6.79288055299980896065481533675, 7.55971591370878823469256548776, 7.74131100109873532120571409940, 8.497014919566527020118211833463, 8.700309598822903808568055572210, 9.269548033660763706910973034590, 9.510415366877337692324840405230, 9.938850994382666261086169202310, 10.36963481193907933889176008593, 10.86878081460772726630137738357, 11.11878951294201265363620957806

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