Properties

Label 4-30e4-1.1-c2e2-0-3
Degree $4$
Conductor $810000$
Sign $1$
Analytic cond. $601.388$
Root an. cond. $4.95209$
Motivic weight $2$
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  + 2·2-s − 8·8-s − 36·13-s − 16·16-s + 20·17-s − 72·26-s + 72·29-s + 40·34-s − 108·37-s − 36·41-s − 10·49-s − 52·53-s + 144·58-s − 148·61-s + 64·64-s − 72·73-s − 216·74-s − 72·82-s + 36·89-s + 144·97-s − 20·98-s − 72·101-s + 288·104-s − 104·106-s − 52·109-s − 20·113-s + 134·121-s + ⋯
L(s)  = 1  + 2-s − 8-s − 2.76·13-s − 16-s + 1.17·17-s − 2.76·26-s + 2.48·29-s + 1.17·34-s − 2.91·37-s − 0.878·41-s − 0.204·49-s − 0.981·53-s + 2.48·58-s − 2.42·61-s + 64-s − 0.986·73-s − 2.91·74-s − 0.878·82-s + 0.404·89-s + 1.48·97-s − 0.204·98-s − 0.712·101-s + 2.76·104-s − 0.981·106-s − 0.477·109-s − 0.176·113-s + 1.10·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(810000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(601.388\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{900} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 810000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.010387414\)
\(L(\frac12)\) \(\approx\) \(1.010387414\)
\(L(2)\) 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$C_2$ \( 1 - p T + p^{2} T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 10 T^{2} + p^{4} T^{4} \)
11$C_2^2$ \( 1 - 134 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 530 T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - 1010 T^{2} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - 36 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 1874 T^{2} + p^{4} T^{4} \)
37$C_2$ \( ( 1 + 54 T + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 3266 T^{2} + p^{4} T^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_2$ \( ( 1 + 26 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 5990 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 74 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 7250 T^{2} + p^{4} T^{4} \)
71$C_2^2$ \( 1 + 718 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 36 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 4370 T^{2} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 5666 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 72 T + p^{2} T^{2} )^{2} \)
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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

−10.23089801128313510943441740399, −9.801281212016367616404375153480, −9.387976789205114897970052659484, −8.740590340222497032252455340999, −8.628667826421034459072840981055, −7.81409981894127600642898685582, −7.63145445143976453830458523809, −6.99910128620688543507556032620, −6.71899796186069682518214152659, −6.16778990892014727661048587078, −5.60758832718049720137127617911, −5.06793166461503951398854127958, −4.75263145445140604838458051010, −4.69765888267163942743976470368, −3.78005164997067597649200975304, −3.14069843552697171708698947382, −2.93957805555843669080057086622, −2.25712449346531607530462782551, −1.44138269675064367895320038094, −0.27263540512954913450864968701, 0.27263540512954913450864968701, 1.44138269675064367895320038094, 2.25712449346531607530462782551, 2.93957805555843669080057086622, 3.14069843552697171708698947382, 3.78005164997067597649200975304, 4.69765888267163942743976470368, 4.75263145445140604838458051010, 5.06793166461503951398854127958, 5.60758832718049720137127617911, 6.16778990892014727661048587078, 6.71899796186069682518214152659, 6.99910128620688543507556032620, 7.63145445143976453830458523809, 7.81409981894127600642898685582, 8.628667826421034459072840981055, 8.740590340222497032252455340999, 9.387976789205114897970052659484, 9.801281212016367616404375153480, 10.23089801128313510943441740399

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