Properties

Label 4-588e2-1.1-c5e2-0-13
Degree $4$
Conductor $345744$
Sign $1$
Analytic cond. $8893.56$
Root an. cond. $9.71111$
Motivic weight $5$
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  + 18·3-s + 47·5-s + 243·9-s − 407·11-s + 449·13-s + 846·15-s − 1.86e3·17-s − 1.46e3·19-s − 44·23-s − 2.82e3·25-s + 2.91e3·27-s + 767·29-s − 1.11e4·31-s − 7.32e3·33-s − 3.11e3·37-s + 8.08e3·39-s − 7.84e3·41-s − 1.26e4·43-s + 1.14e4·45-s + 9.57e3·47-s − 3.36e4·51-s + 1.33e4·53-s − 1.91e4·55-s − 2.63e4·57-s − 4.75e4·59-s − 6.36e4·61-s + 2.11e4·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.840·5-s + 9-s − 1.01·11-s + 0.736·13-s + 0.970·15-s − 1.56·17-s − 0.929·19-s − 0.0173·23-s − 0.903·25-s + 0.769·27-s + 0.169·29-s − 2.08·31-s − 1.17·33-s − 0.373·37-s + 0.850·39-s − 0.728·41-s − 1.04·43-s + 0.840·45-s + 0.632·47-s − 1.81·51-s + 0.655·53-s − 0.852·55-s − 1.07·57-s − 1.77·59-s − 2.19·61-s + 0.619·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(345744\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(8893.56\)
Root analytic conductor: \(9.71111\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 345744,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - p^{2} T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 - 47 T + 5032 T^{2} - 47 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 37 p T + 361744 T^{2} + 37 p^{6} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 449 T + 748730 T^{2} - 449 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 1868 T + 3683746 T^{2} + 1868 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 77 p T + 882870 T^{2} + 77 p^{6} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 44 T - 4829330 T^{2} + 44 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 767 T + 20137030 T^{2} - 767 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 11170 T + 78225563 T^{2} + 11170 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 3113 T + 140329926 T^{2} + 3113 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 7842 T + 247022914 T^{2} + 7842 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 12629 T + 333109116 T^{2} + 12629 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 9576 T + 479320714 T^{2} - 9576 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 13395 T + 786785182 T^{2} - 13395 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 47521 T + 1225624018 T^{2} + 47521 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 63652 T + 2359366478 T^{2} + 63652 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 44541 T + 3121830628 T^{2} - 44541 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 125840 T + 7124822602 T^{2} + 125840 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 6039 T + 3156837796 T^{2} - 6039 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 17588 T + 1504957625 T^{2} - 17588 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 39325 T + 1663856032 T^{2} - 39325 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 83082 T + 12893200018 T^{2} - 83082 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 1905 p T + 241552312 p T^{2} - 1905 p^{6} T^{3} + p^{10} T^{4} \)
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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

−9.402108979184850373613484773864, −9.322367494442466004809934437308, −8.689590071441666485042366219094, −8.671521369871877751444029150487, −7.898167524291616087080901357767, −7.61964196192205606873634900900, −7.07432798677968991427662973035, −6.55529239654547634993881089896, −6.00914371027962963478502220060, −5.74868190524297496600881490007, −4.78133924744764873514058081633, −4.73019311604486794483657969696, −3.70060304537933683646842776416, −3.63558930581180187225155524084, −2.70351092046861496668028830351, −2.31172643502052337788849761468, −1.78400884788701691630491324446, −1.44267253025453914559143419610, 0, 0, 1.44267253025453914559143419610, 1.78400884788701691630491324446, 2.31172643502052337788849761468, 2.70351092046861496668028830351, 3.63558930581180187225155524084, 3.70060304537933683646842776416, 4.73019311604486794483657969696, 4.78133924744764873514058081633, 5.74868190524297496600881490007, 6.00914371027962963478502220060, 6.55529239654547634993881089896, 7.07432798677968991427662973035, 7.61964196192205606873634900900, 7.898167524291616087080901357767, 8.671521369871877751444029150487, 8.689590071441666485042366219094, 9.322367494442466004809934437308, 9.402108979184850373613484773864

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