Properties

Degree $4$
Conductor $1$
Sign $1$
Motivic weight $33$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.21e5·2-s + 3.79e7·3-s + 6.13e9·4-s − 1.81e11·5-s − 4.61e12·6-s − 6.71e13·7-s − 7.37e14·8-s − 8.85e15·9-s + 2.20e16·10-s + 1.33e17·11-s + 2.32e17·12-s − 2.98e18·13-s + 8.17e18·14-s − 6.86e18·15-s + 8.99e19·16-s − 7.93e19·17-s + 1.07e21·18-s − 1.36e21·19-s − 1.11e21·20-s − 2.54e21·21-s − 1.62e22·22-s + 2.61e22·23-s − 2.79e22·24-s − 1.95e23·25-s + 3.62e23·26-s − 5.14e23·27-s − 4.12e23·28-s + ⋯
L(s)  = 1  − 1.31·2-s + 0.508·3-s + 0.714·4-s − 0.530·5-s − 0.667·6-s − 0.763·7-s − 0.926·8-s − 1.59·9-s + 0.696·10-s + 0.878·11-s + 0.363·12-s − 1.24·13-s + 1.00·14-s − 0.269·15-s + 1.21·16-s − 0.395·17-s + 2.09·18-s − 1.08·19-s − 0.379·20-s − 0.388·21-s − 1.15·22-s + 0.889·23-s − 0.471·24-s − 1.68·25-s + 1.63·26-s − 1.24·27-s − 0.545·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(34-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+33/2)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1\)
Sign: $1$
Motivic weight: \(33\)
Character: $\chi_{1} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1,\ (\ :33/2, 33/2),\ 1)\)

Particular Values

\(L(17)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{35}{2})\) 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)$
good2$D_{4}$ \( 1 + 7605 p^{4} T + 2115925 p^{12} T^{2} + 7605 p^{37} T^{3} + p^{66} T^{4} \)
3$D_{4}$ \( 1 - 1404440 p^{3} T + 522714408050 p^{9} T^{2} - 1404440 p^{36} T^{3} + p^{66} T^{4} \)
5$D_{4}$ \( 1 + 1448492292 p^{3} T + 585507855025144558 p^{8} T^{2} + 1448492292 p^{36} T^{3} + p^{66} T^{4} \)
7$D_{4}$ \( 1 + 9593297152400 p T + \)\(36\!\cdots\!50\)\( p^{4} T^{2} + 9593297152400 p^{34} T^{3} + p^{66} T^{4} \)
11$D_{4}$ \( 1 - 12170165040174024 p T + \)\(23\!\cdots\!66\)\( p^{2} T^{2} - 12170165040174024 p^{34} T^{3} + p^{66} T^{4} \)
13$D_{4}$ \( 1 + 229354652173803380 p T + \)\(61\!\cdots\!50\)\( p^{3} T^{2} + 229354652173803380 p^{34} T^{3} + p^{66} T^{4} \)
17$D_{4}$ \( 1 + 79361149261175525340 T - \)\(26\!\cdots\!50\)\( p T^{2} + 79361149261175525340 p^{33} T^{3} + p^{66} T^{4} \)
19$D_{4}$ \( 1 + \)\(13\!\cdots\!00\)\( T + \)\(17\!\cdots\!22\)\( p T^{2} + \)\(13\!\cdots\!00\)\( p^{33} T^{3} + p^{66} T^{4} \)
23$D_{4}$ \( 1 - \)\(26\!\cdots\!40\)\( T + \)\(68\!\cdots\!50\)\( p T^{2} - \)\(26\!\cdots\!40\)\( p^{33} T^{3} + p^{66} T^{4} \)
29$D_{4}$ \( 1 + \)\(57\!\cdots\!00\)\( p T + \)\(46\!\cdots\!58\)\( p^{2} T^{2} + \)\(57\!\cdots\!00\)\( p^{34} T^{3} + p^{66} T^{4} \)
31$D_{4}$ \( 1 + \)\(20\!\cdots\!36\)\( p T + \)\(30\!\cdots\!86\)\( p^{2} T^{2} + \)\(20\!\cdots\!36\)\( p^{34} T^{3} + p^{66} T^{4} \)
37$D_{4}$ \( 1 + \)\(10\!\cdots\!20\)\( T + \)\(13\!\cdots\!50\)\( T^{2} + \)\(10\!\cdots\!20\)\( p^{33} T^{3} + p^{66} T^{4} \)
41$D_{4}$ \( 1 - \)\(27\!\cdots\!44\)\( T + \)\(22\!\cdots\!26\)\( T^{2} - \)\(27\!\cdots\!44\)\( p^{33} T^{3} + p^{66} T^{4} \)
43$D_{4}$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(12\!\cdots\!50\)\( T^{2} - \)\(15\!\cdots\!00\)\( p^{33} T^{3} + p^{66} T^{4} \)
47$D_{4}$ \( 1 - \)\(11\!\cdots\!20\)\( p T + \)\(37\!\cdots\!50\)\( T^{2} - \)\(11\!\cdots\!20\)\( p^{34} T^{3} + p^{66} T^{4} \)
53$D_{4}$ \( 1 + \)\(26\!\cdots\!20\)\( T + \)\(12\!\cdots\!50\)\( T^{2} + \)\(26\!\cdots\!20\)\( p^{33} T^{3} + p^{66} T^{4} \)
59$D_{4}$ \( 1 + \)\(30\!\cdots\!00\)\( T + \)\(77\!\cdots\!58\)\( T^{2} + \)\(30\!\cdots\!00\)\( p^{33} T^{3} + p^{66} T^{4} \)
61$D_{4}$ \( 1 + \)\(57\!\cdots\!36\)\( T + \)\(16\!\cdots\!86\)\( T^{2} + \)\(57\!\cdots\!36\)\( p^{33} T^{3} + p^{66} T^{4} \)
67$D_{4}$ \( 1 - \)\(15\!\cdots\!60\)\( T + \)\(41\!\cdots\!50\)\( T^{2} - \)\(15\!\cdots\!60\)\( p^{33} T^{3} + p^{66} T^{4} \)
71$D_{4}$ \( 1 + \)\(26\!\cdots\!76\)\( T + \)\(22\!\cdots\!66\)\( T^{2} + \)\(26\!\cdots\!76\)\( p^{33} T^{3} + p^{66} T^{4} \)
73$D_{4}$ \( 1 - \)\(94\!\cdots\!40\)\( T + \)\(19\!\cdots\!50\)\( T^{2} - \)\(94\!\cdots\!40\)\( p^{33} T^{3} + p^{66} T^{4} \)
79$D_{4}$ \( 1 + \)\(85\!\cdots\!00\)\( T + \)\(85\!\cdots\!78\)\( T^{2} + \)\(85\!\cdots\!00\)\( p^{33} T^{3} + p^{66} T^{4} \)
83$D_{4}$ \( 1 - \)\(29\!\cdots\!20\)\( T + \)\(37\!\cdots\!50\)\( T^{2} - \)\(29\!\cdots\!20\)\( p^{33} T^{3} + p^{66} T^{4} \)
89$D_{4}$ \( 1 - \)\(13\!\cdots\!00\)\( T + \)\(47\!\cdots\!38\)\( T^{2} - \)\(13\!\cdots\!00\)\( p^{33} T^{3} + p^{66} T^{4} \)
97$D_{4}$ \( 1 + \)\(36\!\cdots\!60\)\( T + \)\(42\!\cdots\!50\)\( T^{2} + \)\(36\!\cdots\!60\)\( p^{33} T^{3} + p^{66} T^{4} \)
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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

−24.53206884006730080481924156153, −23.17164710223873043569343353110, −22.08212852190947044930996024239, −20.47185464506878550620095650151, −19.37737814171362848300641710869, −19.25851336377166860001110437885, −17.42233318430714526366111398424, −16.99729322981038600458338655937, −15.31150255031509647696477568593, −14.33154534246216629846064850558, −12.38824383391611598453844590283, −11.16259272438525703707451648647, −9.337420969687700231085444375144, −8.942750425824562441874080636099, −7.58479795202813569245751961691, −5.97786009657894493138722886448, −3.56400685368044213449625106689, −2.31107464957568549956618907631, 0, 0, 2.31107464957568549956618907631, 3.56400685368044213449625106689, 5.97786009657894493138722886448, 7.58479795202813569245751961691, 8.942750425824562441874080636099, 9.337420969687700231085444375144, 11.16259272438525703707451648647, 12.38824383391611598453844590283, 14.33154534246216629846064850558, 15.31150255031509647696477568593, 16.99729322981038600458338655937, 17.42233318430714526366111398424, 19.25851336377166860001110437885, 19.37737814171362848300641710869, 20.47185464506878550620095650151, 22.08212852190947044930996024239, 23.17164710223873043569343353110, 24.53206884006730080481924156153

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