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Properties

Label 4-123e2-1.1-c1e2-0-0
Degree $4$
Conductor $15129$
Sign $1$
Analytic cond. $0.964638$
Root an. cond. $0.991039$
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  + 2·3-s − 2·4-s + 4·5-s − 4·7-s + 3·9-s + 2·11-s − 4·12-s + 4·13-s + 8·15-s + 2·17-s − 8·19-s − 8·20-s − 8·21-s + 4·25-s + 4·27-s + 8·28-s + 2·29-s − 6·31-s + 4·33-s − 16·35-s − 6·36-s − 2·37-s + 8·39-s − 2·41-s − 10·43-s − 4·44-s + 12·45-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s + 1.78·5-s − 1.51·7-s + 9-s + 0.603·11-s − 1.15·12-s + 1.10·13-s + 2.06·15-s + 0.485·17-s − 1.83·19-s − 1.78·20-s − 1.74·21-s + 4/5·25-s + 0.769·27-s + 1.51·28-s + 0.371·29-s − 1.07·31-s + 0.696·33-s − 2.70·35-s − 36-s − 0.328·37-s + 1.28·39-s − 0.312·41-s − 1.52·43-s − 0.603·44-s + 1.78·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15129\)    =    \(3^{2} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(0.964638\)
Root analytic conductor: \(0.991039\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15129,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.481058246\)
\(L(\frac12)\) \(\approx\) \(1.481058246\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
41$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 18 T + 173 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T + 91 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 6 T + 101 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
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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

−13.46399897906393686856027095324, −13.45695470008391059890228053777, −12.81844717494374397842697575262, −12.71163363372123667207009067189, −11.77294837495192627806218510543, −10.78347769354339306901080134615, −10.18278166973852136003053300517, −9.994795573797464625637559537776, −9.258714987889494237299891927528, −9.121296601361451419041207696796, −8.663540701558035472144006964402, −8.065991432819730803053101151210, −6.86136765449303973440793018188, −6.58052100325357270844172164728, −5.92263008191291302094574962741, −5.31085207599506180561686349009, −4.06319251959605651767610147604, −3.76990265978882257353141619058, −2.70227925347984232972683592642, −1.77891449336676596242213201754, 1.77891449336676596242213201754, 2.70227925347984232972683592642, 3.76990265978882257353141619058, 4.06319251959605651767610147604, 5.31085207599506180561686349009, 5.92263008191291302094574962741, 6.58052100325357270844172164728, 6.86136765449303973440793018188, 8.065991432819730803053101151210, 8.663540701558035472144006964402, 9.121296601361451419041207696796, 9.258714987889494237299891927528, 9.994795573797464625637559537776, 10.18278166973852136003053300517, 10.78347769354339306901080134615, 11.77294837495192627806218510543, 12.71163363372123667207009067189, 12.81844717494374397842697575262, 13.45695470008391059890228053777, 13.46399897906393686856027095324

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