Properties

Label 4-33e4-1.1-c3e2-0-3
Degree $4$
Conductor $1185921$
Sign $1$
Analytic cond. $4128.45$
Root an. cond. $8.01580$
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  − 2·2-s − 4-s + 10·5-s + 8·7-s − 4·8-s − 20·10-s − 130·13-s − 16·14-s − 19·16-s − 14·17-s + 48·19-s − 10·20-s + 128·23-s − 163·25-s + 260·26-s − 8·28-s − 30·29-s − 184·31-s + 202·32-s + 28·34-s + 80·35-s + 126·37-s − 96·38-s − 40·40-s + 370·41-s + 264·43-s − 256·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/8·4-s + 0.894·5-s + 0.431·7-s − 0.176·8-s − 0.632·10-s − 2.77·13-s − 0.305·14-s − 0.296·16-s − 0.199·17-s + 0.579·19-s − 0.111·20-s + 1.16·23-s − 1.30·25-s + 1.96·26-s − 0.0539·28-s − 0.192·29-s − 1.06·31-s + 1.11·32-s + 0.141·34-s + 0.386·35-s + 0.559·37-s − 0.409·38-s − 0.158·40-s + 1.40·41-s + 0.936·43-s − 0.820·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1185921\)    =    \(3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4128.45\)
Root analytic conductor: \(8.01580\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1185921,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.393712708\)
\(L(\frac12)\) \(\approx\) \(1.393712708\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
11 \( 1 \)
good2$D_{4}$ \( 1 + p T + 5 T^{2} + p^{4} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - 2 p T + 263 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 8 T + 114 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 10 p T + 8607 T^{2} + 10 p^{4} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 14 T + 5987 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 48 T + 13322 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 128 T + 15362 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 30 T + 32575 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 184 T + 41538 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 126 T + 30383 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 370 T + 172019 T^{2} - 370 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 264 T + 170630 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 256 T + 76178 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 162 T + 217615 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 1304 T + 814694 T^{2} - 1304 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 300 T + 374894 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 656 T + 707658 T^{2} + 656 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 1176 T + 1023934 T^{2} - 1176 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 668 T + 500790 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 416 T + 886770 T^{2} + 416 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 960 T + 1342762 T^{2} + 960 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1074 T + 1326595 T^{2} - 1074 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 338 T + 1196835 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.537691773393255577006059494258, −9.517046589720918422743302160166, −9.022530588374038581423778626165, −8.613988810527275089177050912487, −7.88723344479294634915884856093, −7.79258622352720849562537900159, −7.16579890593403550827398929611, −6.99866176008144604398226026610, −6.43210255016540815738082777003, −5.65519685650764544417367188193, −5.52231453465964332953839396420, −5.04985599893705187566762006323, −4.53998703557849889106836395315, −4.12397690063656553624817031925, −3.32561571731497337920574766434, −2.60142248845277907894792962417, −2.25781248779417798522424691039, −1.93623153677733689207418339686, −0.864761260769871819672149775817, −0.40598454879212157604919643259, 0.40598454879212157604919643259, 0.864761260769871819672149775817, 1.93623153677733689207418339686, 2.25781248779417798522424691039, 2.60142248845277907894792962417, 3.32561571731497337920574766434, 4.12397690063656553624817031925, 4.53998703557849889106836395315, 5.04985599893705187566762006323, 5.52231453465964332953839396420, 5.65519685650764544417367188193, 6.43210255016540815738082777003, 6.99866176008144604398226026610, 7.16579890593403550827398929611, 7.79258622352720849562537900159, 7.88723344479294634915884856093, 8.613988810527275089177050912487, 9.022530588374038581423778626165, 9.517046589720918422743302160166, 9.537691773393255577006059494258

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