Properties

Label 4-45e2-1.1-c27e2-0-0
Degree $4$
Conductor $2025$
Sign $1$
Analytic cond. $43195.3$
Root an. cond. $14.4164$
Motivic weight $27$
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  + 1.37e8·4-s + 9.81e14·16-s + 2.19e17·19-s − 7.45e18·25-s − 1.86e20·31-s + 1.31e23·49-s + 1.61e24·61-s − 2.34e24·64-s + 3.02e25·76-s − 1.53e26·79-s − 1.02e27·100-s − 1.07e28·109-s − 2.62e28·121-s − 2.56e28·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.38e30·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 1.02·4-s + 0.0544·16-s + 1.19·19-s − 25-s − 1.37·31-s + 2·49-s + 1.27·61-s − 0.970·64-s + 1.22·76-s − 3.69·79-s − 1.02·100-s − 3.34·109-s − 2·121-s − 1.40·124-s + 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(43195.3\)
Root analytic conductor: \(14.4164\)
Motivic weight: \(27\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2025,\ (\ :27/2, 27/2),\ 1)\)

Particular Values

\(L(14)\) \(\approx\) \(0.7348433960\)
\(L(\frac12)\) \(\approx\) \(0.7348433960\)
\(L(\frac{29}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + p^{27} T^{2} \)
good2$C_2^2$ \( 1 - 137823851 T^{2} + p^{54} T^{4} \)
7$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
11$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
13$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
17$C_2^2$ \( 1 - \)\(29\!\cdots\!26\)\( T^{2} + p^{54} T^{4} \)
19$C_2$ \( ( 1 - 109733178357376004 T + p^{27} T^{2} )^{2} \)
23$C_2^2$ \( 1 + \)\(52\!\cdots\!86\)\( T^{2} + p^{54} T^{4} \)
29$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 93193506284515571752 T + p^{27} T^{2} )^{2} \)
37$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
43$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
47$C_2^2$ \( 1 - \)\(21\!\cdots\!46\)\( T^{2} + p^{54} T^{4} \)
53$C_2^2$ \( 1 + \)\(29\!\cdots\!46\)\( T^{2} + p^{54} T^{4} \)
59$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
61$C_2$ \( ( 1 - \)\(80\!\cdots\!82\)\( T + p^{27} T^{2} )^{2} \)
67$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
71$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
73$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
79$C_2$ \( ( 1 + \)\(76\!\cdots\!64\)\( T + p^{27} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(80\!\cdots\!34\)\( T^{2} + p^{54} T^{4} \)
89$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
97$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
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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

−11.22635988447275399282232813563, −10.49029826488493496936944323582, −10.18277650680437817899807132702, −9.375365224945746317106420095135, −9.033724963088822378323457485365, −8.257621796461604555474489953675, −7.60301532212992527815415357146, −7.23467243532439341668021457434, −6.80246784577583636968307548890, −6.08658220962331645165265902348, −5.52929081497347232929272021447, −5.21688236131505500082286747159, −4.17429625080559402087087288564, −3.86527519619025042834763129456, −3.11088083300293128658690927042, −2.54733202720512859947311374051, −2.17815853238366495466063728376, −1.37938261532752268918654850577, −1.16605221930568565540667355031, −0.14522626889455851092667612719, 0.14522626889455851092667612719, 1.16605221930568565540667355031, 1.37938261532752268918654850577, 2.17815853238366495466063728376, 2.54733202720512859947311374051, 3.11088083300293128658690927042, 3.86527519619025042834763129456, 4.17429625080559402087087288564, 5.21688236131505500082286747159, 5.52929081497347232929272021447, 6.08658220962331645165265902348, 6.80246784577583636968307548890, 7.23467243532439341668021457434, 7.60301532212992527815415357146, 8.257621796461604555474489953675, 9.033724963088822378323457485365, 9.375365224945746317106420095135, 10.18277650680437817899807132702, 10.49029826488493496936944323582, 11.22635988447275399282232813563

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