Properties

Label 4-45e2-1.1-c19e2-0-0
Degree $4$
Conductor $2025$
Sign $1$
Analytic cond. $10602.3$
Root an. cond. $10.1472$
Motivic weight $19$
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  − 1.04e6·4-s + 8.16e11·16-s + 2.02e12·19-s − 1.90e13·25-s − 2.71e14·31-s + 2.27e16·49-s − 2.35e17·61-s − 5.65e17·64-s − 2.11e18·76-s + 2.78e18·79-s + 1.99e19·100-s − 8.93e19·109-s − 1.22e20·121-s + 2.83e20·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.92e21·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.99·4-s + 2.96·16-s + 1.43·19-s − 25-s − 1.84·31-s + 2·49-s − 2.58·61-s − 3.92·64-s − 2.86·76-s + 2.61·79-s + 1.99·100-s − 3.93·109-s − 2·121-s + 3.67·124-s + 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(10602.3\)
Root analytic conductor: \(10.1472\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2025,\ (\ :19/2, 19/2),\ 1)\)

Particular Values

\(L(10)\) \(\approx\) \(0.01126516604\)
\(L(\frac12)\) \(\approx\) \(0.01126516604\)
\(L(\frac{21}{2})\) 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$C_2$ \( 1 + p^{19} T^{2} \)
good2$C_2^2$ \( 1 + 1044469 T^{2} + p^{38} T^{4} \)
7$C_2$ \( ( 1 - p^{19} T^{2} )^{2} \)
11$C_2$ \( ( 1 + p^{19} T^{2} )^{2} \)
13$C_2$ \( ( 1 - p^{19} T^{2} )^{2} \)
17$C_2^2$ \( 1 + \)\(46\!\cdots\!74\)\( T^{2} + p^{38} T^{4} \)
19$C_2$ \( ( 1 - 1011668819684 T + p^{19} T^{2} )^{2} \)
23$C_2^2$ \( 1 + \)\(14\!\cdots\!46\)\( T^{2} + p^{38} T^{4} \)
29$C_2$ \( ( 1 + p^{19} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 135667566768232 T + p^{19} T^{2} )^{2} \)
37$C_2$ \( ( 1 - p^{19} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{19} T^{2} )^{2} \)
43$C_2$ \( ( 1 - p^{19} T^{2} )^{2} \)
47$C_2^2$ \( 1 + \)\(34\!\cdots\!54\)\( T^{2} + p^{38} T^{4} \)
53$C_2^2$ \( 1 - \)\(88\!\cdots\!54\)\( T^{2} + p^{38} T^{4} \)
59$C_2$ \( ( 1 + p^{19} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 117984697177297318 T + p^{19} T^{2} )^{2} \)
67$C_2$ \( ( 1 - p^{19} T^{2} )^{2} \)
71$C_2$ \( ( 1 + p^{19} T^{2} )^{2} \)
73$C_2$ \( ( 1 - p^{19} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 1392724254857414096 T + p^{19} T^{2} )^{2} \)
83$C_2^2$ \( 1 + \)\(32\!\cdots\!86\)\( T^{2} + p^{38} T^{4} \)
89$C_2$ \( ( 1 + p^{19} T^{2} )^{2} \)
97$C_2$ \( ( 1 - p^{19} T^{2} )^{2} \)
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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

−12.06399643993577294024851530462, −12.03847302638364941770750945898, −10.82403139593092634467484417920, −10.44513870072371749802783323048, −9.627210920249074714592923112224, −9.189984810032066003510411765591, −9.072203024984915142053365821848, −7.972557298630044624981059695185, −7.83382921589307291526852907998, −7.03697163821763924828839826535, −5.91151523494885755471820058736, −5.51632230010678617389142658170, −5.01247577509519068357022516195, −4.33370821762938093286939300965, −3.70050311568136649932398201406, −3.39064754366464078725865353173, −2.39684031760678589903251986273, −1.37501392821846797412155302773, −0.999918994205370329983526860416, −0.02885685102828425857186272393, 0.02885685102828425857186272393, 0.999918994205370329983526860416, 1.37501392821846797412155302773, 2.39684031760678589903251986273, 3.39064754366464078725865353173, 3.70050311568136649932398201406, 4.33370821762938093286939300965, 5.01247577509519068357022516195, 5.51632230010678617389142658170, 5.91151523494885755471820058736, 7.03697163821763924828839826535, 7.83382921589307291526852907998, 7.972557298630044624981059695185, 9.072203024984915142053365821848, 9.189984810032066003510411765591, 9.627210920249074714592923112224, 10.44513870072371749802783323048, 10.82403139593092634467484417920, 12.03847302638364941770750945898, 12.06399643993577294024851530462

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