Properties

Label 4-30e4-1.1-c3e2-0-5
Degree $4$
Conductor $810000$
Sign $1$
Analytic cond. $2819.79$
Root an. cond. $7.28709$
Motivic weight $3$
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  − 60·11-s + 56·19-s + 420·29-s − 8·31-s − 480·41-s + 682·49-s − 900·59-s − 332·61-s + 2.04e3·71-s + 1.83e3·79-s − 840·89-s − 900·101-s + 3.40e3·109-s + 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.37e3·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.64·11-s + 0.676·19-s + 2.68·29-s − 0.0463·31-s − 1.82·41-s + 1.98·49-s − 1.98·59-s − 0.696·61-s + 3.40·71-s + 2.60·79-s − 1.00·89-s − 0.886·101-s + 2.99·109-s + 0.0285·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.99·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(810000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2819.79\)
Root analytic conductor: \(7.28709\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 810000,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.397536694\)
\(L(\frac12)\) \(\approx\) \(2.397536694\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 682 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 30 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 4378 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 1726 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 28 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 9934 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 210 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 61306 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 240 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 140518 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 193246 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 296854 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 + 450 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 166 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 222938 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 1020 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 715534 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 916 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 156026 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 420 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 + 540098 T^{2} + p^{6} T^{4} \)
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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

−9.859958741930050112163421641324, −9.669766547832549444602307137189, −9.108312963500059114962368417645, −8.560655716411356922320459271382, −8.118443000210158932411939371525, −8.045841514149026019984096396970, −7.39385929967941870811973318823, −6.99478329200163005348377374548, −6.48105692284846780380498347276, −6.12306024057428451509262079667, −5.36829310564883795972252311327, −5.17244807287499468475779863356, −4.72278427671724149084418650943, −4.20270811482181738308586973216, −3.35207164519829761499528186795, −3.08247278493990379848327026725, −2.45856659559515618628989173959, −1.94750423049145117134403054181, −0.984316828859506754651978103772, −0.47272824774670665840605765436, 0.47272824774670665840605765436, 0.984316828859506754651978103772, 1.94750423049145117134403054181, 2.45856659559515618628989173959, 3.08247278493990379848327026725, 3.35207164519829761499528186795, 4.20270811482181738308586973216, 4.72278427671724149084418650943, 5.17244807287499468475779863356, 5.36829310564883795972252311327, 6.12306024057428451509262079667, 6.48105692284846780380498347276, 6.99478329200163005348377374548, 7.39385929967941870811973318823, 8.045841514149026019984096396970, 8.118443000210158932411939371525, 8.560655716411356922320459271382, 9.108312963500059114962368417645, 9.669766547832549444602307137189, 9.859958741930050112163421641324

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