Properties

Label 4-45e2-1.1-c21e2-0-0
Degree $4$
Conductor $2025$
Sign $1$
Analytic cond. $15816.7$
Root an. cond. $11.2144$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.47e6·4-s + 7.68e12·16-s + 5.90e13·19-s − 4.76e14·25-s − 1.64e16·31-s + 1.11e18·49-s + 9.60e18·61-s + 1.14e19·64-s + 2.05e20·76-s + 3.36e20·79-s − 1.65e21·100-s + 2.69e21·109-s − 1.48e22·121-s − 5.72e22·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.94e23·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 1.65·4-s + 1.74·16-s + 2.21·19-s − 25-s − 3.60·31-s + 2·49-s + 1.72·61-s + 1.23·64-s + 3.66·76-s + 3.99·79-s − 1.65·100-s + 1.09·109-s − 2·121-s − 5.97·124-s + 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(15816.7\)
Root analytic conductor: \(11.2144\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2025,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(6.087622547\)
\(L(\frac12)\) \(\approx\) \(6.087622547\)
\(L(\frac{23}{2})\) 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$C_2$ \( 1 + p^{21} T^{2} \)
good2$C_2^2$ \( 1 - 3476099 T^{2} + p^{42} T^{4} \)
7$C_2$ \( ( 1 - p^{21} T^{2} )^{2} \)
11$C_2$ \( ( 1 + p^{21} T^{2} )^{2} \)
13$C_2$ \( ( 1 - p^{21} T^{2} )^{2} \)
17$C_2^2$ \( 1 - \)\(67\!\cdots\!14\)\( T^{2} + p^{42} T^{4} \)
19$C_2$ \( ( 1 - 29536728219596 T + p^{21} T^{2} )^{2} \)
23$C_2^2$ \( 1 - \)\(77\!\cdots\!66\)\( T^{2} + p^{42} T^{4} \)
29$C_2$ \( ( 1 + p^{21} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8231708685659992 T + p^{21} T^{2} )^{2} \)
37$C_2$ \( ( 1 - p^{21} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{21} T^{2} )^{2} \)
43$C_2$ \( ( 1 - p^{21} T^{2} )^{2} \)
47$C_2^2$ \( 1 - \)\(58\!\cdots\!14\)\( T^{2} + p^{42} T^{4} \)
53$C_2^2$ \( 1 - \)\(27\!\cdots\!86\)\( T^{2} + p^{42} T^{4} \)
59$C_2$ \( ( 1 + p^{21} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4800688498498618802 T + p^{21} T^{2} )^{2} \)
67$C_2$ \( ( 1 - p^{21} T^{2} )^{2} \)
71$C_2$ \( ( 1 + p^{21} T^{2} )^{2} \)
73$C_2$ \( ( 1 - p^{21} T^{2} )^{2} \)
79$C_2$ \( ( 1 - \)\(16\!\cdots\!84\)\( T + p^{21} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(71\!\cdots\!46\)\( T^{2} + p^{42} T^{4} \)
89$C_2$ \( ( 1 + p^{21} T^{2} )^{2} \)
97$C_2$ \( ( 1 - p^{21} T^{2} )^{2} \)
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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

−11.94990989411473051022532805445, −11.30328752233750272810599382058, −10.84880603976909862598727868849, −10.41308552491636412337055058240, −9.402209379582557831096666659781, −9.387493847404681578983761753043, −8.252103102674108818351787488563, −7.53133524794276029166181851641, −7.32237743766892152777113978686, −6.83907115468086695498348821913, −5.97189388587692062459713270149, −5.52454964221300856022833526358, −5.13314027105602053047432896454, −3.71959911526629136307525644024, −3.65174765729432043217799744428, −2.84700106178930892045948771056, −2.08811921860502155089513970832, −1.87499761754620523689147830506, −1.06680097203111146941919170743, −0.50738432469970482956404683227, 0.50738432469970482956404683227, 1.06680097203111146941919170743, 1.87499761754620523689147830506, 2.08811921860502155089513970832, 2.84700106178930892045948771056, 3.65174765729432043217799744428, 3.71959911526629136307525644024, 5.13314027105602053047432896454, 5.52454964221300856022833526358, 5.97189388587692062459713270149, 6.83907115468086695498348821913, 7.32237743766892152777113978686, 7.53133524794276029166181851641, 8.252103102674108818351787488563, 9.387493847404681578983761753043, 9.402209379582557831096666659781, 10.41308552491636412337055058240, 10.84880603976909862598727868849, 11.30328752233750272810599382058, 11.94990989411473051022532805445

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