Properties

Label 4-90e2-1.1-c2e2-0-1
Degree $4$
Conductor $8100$
Sign $1$
Analytic cond. $6.01388$
Root an. cond. $1.56598$
Motivic weight $2$
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  + 2·2-s + 2·4-s + 4·7-s + 16·11-s + 6·13-s + 8·14-s − 4·16-s − 14·17-s + 32·22-s + 4·23-s − 25·25-s + 12·26-s + 8·28-s + 104·31-s − 8·32-s − 28·34-s − 6·37-s + 16·41-s − 84·43-s + 32·44-s + 8·46-s + 36·47-s + 8·49-s − 50·50-s + 12·52-s − 106·53-s − 96·61-s + ⋯
L(s)  = 1  + 2-s + 1/2·4-s + 4/7·7-s + 1.45·11-s + 6/13·13-s + 4/7·14-s − 1/4·16-s − 0.823·17-s + 1.45·22-s + 4/23·23-s − 25-s + 6/13·26-s + 2/7·28-s + 3.35·31-s − 1/4·32-s − 0.823·34-s − 0.162·37-s + 0.390·41-s − 1.95·43-s + 8/11·44-s + 4/23·46-s + 0.765·47-s + 8/49·49-s − 50-s + 3/13·52-s − 2·53-s − 1.57·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(6.01388\)
Root analytic conductor: \(1.56598\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8100,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.830195560\)
\(L(\frac12)\) \(\approx\) \(2.830195560\)
\(L(2)\) 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$C_2$ \( 1 - p T + p T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + p^{2} T^{2} \)
good7$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2$ \( ( 1 - 16 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \)
19$C_2^2$ \( 1 - 322 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \)
31$C_2$ \( ( 1 - 52 T + p^{2} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 84 T + 3528 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \)
59$C_2^2$ \( 1 - 6562 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 48 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 124 T + 7688 T^{2} - 124 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2$ \( ( 1 - 28 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 94 T + 4418 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2^2$ \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 9442 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 + 126 T + 7938 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \)
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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

−14.13431962902395914366131936347, −13.57316499754929552636352085392, −13.35405931161454968575998111711, −12.50323212053148142590062758371, −11.87787309647466466892041219742, −11.74529163801427426859045693301, −11.13697289022499561290794207190, −10.51396192154898461426291912661, −9.683088508157704384785666121113, −9.286388328972080110546009079123, −8.299684347999981198896119438282, −8.192394694420734508389404915960, −7.00971697305070295836234339632, −6.43122587510357414693953954050, −6.07898401272303230111741068109, −4.99702136963147333187798867256, −4.43971308324919386185934448900, −3.82730273436598180181313406219, −2.79394477055932692532199011707, −1.48102558081890001302219738510, 1.48102558081890001302219738510, 2.79394477055932692532199011707, 3.82730273436598180181313406219, 4.43971308324919386185934448900, 4.99702136963147333187798867256, 6.07898401272303230111741068109, 6.43122587510357414693953954050, 7.00971697305070295836234339632, 8.192394694420734508389404915960, 8.299684347999981198896119438282, 9.286388328972080110546009079123, 9.683088508157704384785666121113, 10.51396192154898461426291912661, 11.13697289022499561290794207190, 11.74529163801427426859045693301, 11.87787309647466466892041219742, 12.50323212053148142590062758371, 13.35405931161454968575998111711, 13.57316499754929552636352085392, 14.13431962902395914366131936347

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