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Properties

Label 4-1-1.1-c29e2-0-0
Degree $4$
Conductor $1$
Sign $1$
Analytic cond. $28.3854$
Root an. cond. $2.30820$
Motivic weight $29$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.64e3·2-s − 4.96e6·3-s − 5.44e8·4-s − 1.74e10·5-s − 4.29e10·6-s − 3.02e12·7-s − 5.41e12·8-s + 2.54e13·9-s − 1.51e14·10-s − 2.05e15·11-s + 2.70e15·12-s + 1.71e16·13-s − 2.60e16·14-s + 8.68e16·15-s + 4.22e16·16-s − 6.64e17·17-s + 2.19e17·18-s + 1.23e18·19-s + 9.51e18·20-s + 1.50e19·21-s − 1.77e19·22-s − 1.85e19·23-s + 2.69e19·24-s − 1.37e20·25-s + 1.48e20·26-s − 4.71e20·27-s + 1.64e21·28-s + ⋯
L(s)  = 1  + 0.372·2-s − 0.599·3-s − 1.01·4-s − 1.28·5-s − 0.223·6-s − 1.68·7-s − 0.435·8-s + 0.370·9-s − 0.477·10-s − 1.63·11-s + 0.608·12-s + 1.20·13-s − 0.627·14-s + 0.767·15-s + 0.146·16-s − 0.957·17-s + 0.138·18-s + 0.353·19-s + 1.29·20-s + 1.00·21-s − 0.608·22-s − 0.334·23-s + 0.261·24-s − 0.735·25-s + 0.450·26-s − 0.828·27-s + 1.70·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(30-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+29/2)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(28.3854\)
Root analytic conductor: \(2.30820\)
Motivic weight: \(29\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1,\ (\ :29/2, 29/2),\ 1)\)

Particular Values

\(L(15)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{31}{2})\) 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$D_{4}$ \( 1 - 135 p^{6} T + 151165 p^{12} T^{2} - 135 p^{35} T^{3} + p^{58} T^{4} \)
3$D_{4}$ \( 1 + 551960 p^{2} T - 350106910 p^{7} T^{2} + 551960 p^{31} T^{3} + p^{58} T^{4} \)
5$D_{4}$ \( 1 + 139822308 p^{3} T + 5664377441277062 p^{7} T^{2} + 139822308 p^{32} T^{3} + p^{58} T^{4} \)
7$D_{4}$ \( 1 + 431473240400 p T + \)\(19\!\cdots\!50\)\( p^{3} T^{2} + 431473240400 p^{30} T^{3} + p^{58} T^{4} \)
11$D_{4}$ \( 1 + 186852774508056 p T + \)\(31\!\cdots\!66\)\( p^{3} T^{2} + 186852774508056 p^{30} T^{3} + p^{58} T^{4} \)
13$D_{4}$ \( 1 - 1318413167640940 p T + \)\(21\!\cdots\!30\)\( p^{3} T^{2} - 1318413167640940 p^{30} T^{3} + p^{58} T^{4} \)
17$D_{4}$ \( 1 + 39105642818102940 p T + \)\(31\!\cdots\!30\)\( p^{2} T^{2} + 39105642818102940 p^{30} T^{3} + p^{58} T^{4} \)
19$D_{4}$ \( 1 - 1232169445452155080 T + \)\(78\!\cdots\!82\)\( p T^{2} - 1232169445452155080 p^{29} T^{3} + p^{58} T^{4} \)
23$D_{4}$ \( 1 + 808192630624973040 p T + \)\(11\!\cdots\!10\)\( p^{2} T^{2} + 808192630624973040 p^{30} T^{3} + p^{58} T^{4} \)
29$D_{4}$ \( 1 - \)\(99\!\cdots\!20\)\( T + \)\(34\!\cdots\!38\)\( T^{2} - \)\(99\!\cdots\!20\)\( p^{29} T^{3} + p^{58} T^{4} \)
31$D_{4}$ \( 1 + \)\(10\!\cdots\!56\)\( T + \)\(17\!\cdots\!26\)\( T^{2} + \)\(10\!\cdots\!56\)\( p^{29} T^{3} + p^{58} T^{4} \)
37$D_{4}$ \( 1 - \)\(98\!\cdots\!60\)\( T + \)\(75\!\cdots\!10\)\( T^{2} - \)\(98\!\cdots\!60\)\( p^{29} T^{3} + p^{58} T^{4} \)
41$D_{4}$ \( 1 + \)\(10\!\cdots\!76\)\( T + \)\(11\!\cdots\!66\)\( T^{2} + \)\(10\!\cdots\!76\)\( p^{29} T^{3} + p^{58} T^{4} \)
43$D_{4}$ \( 1 - \)\(51\!\cdots\!00\)\( T + \)\(36\!\cdots\!50\)\( T^{2} - \)\(51\!\cdots\!00\)\( p^{29} T^{3} + p^{58} T^{4} \)
47$D_{4}$ \( 1 + \)\(45\!\cdots\!20\)\( T + \)\(11\!\cdots\!30\)\( T^{2} + \)\(45\!\cdots\!20\)\( p^{29} T^{3} + p^{58} T^{4} \)
53$D_{4}$ \( 1 + \)\(16\!\cdots\!40\)\( T + \)\(22\!\cdots\!30\)\( T^{2} + \)\(16\!\cdots\!40\)\( p^{29} T^{3} + p^{58} T^{4} \)
59$D_{4}$ \( 1 + \)\(83\!\cdots\!60\)\( T + \)\(40\!\cdots\!78\)\( T^{2} + \)\(83\!\cdots\!60\)\( p^{29} T^{3} + p^{58} T^{4} \)
61$D_{4}$ \( 1 + \)\(15\!\cdots\!16\)\( T + \)\(90\!\cdots\!46\)\( T^{2} + \)\(15\!\cdots\!16\)\( p^{29} T^{3} + p^{58} T^{4} \)
67$D_{4}$ \( 1 - \)\(24\!\cdots\!20\)\( T + \)\(16\!\cdots\!70\)\( T^{2} - \)\(24\!\cdots\!20\)\( p^{29} T^{3} + p^{58} T^{4} \)
71$D_{4}$ \( 1 + \)\(18\!\cdots\!36\)\( T + \)\(90\!\cdots\!86\)\( T^{2} + \)\(18\!\cdots\!36\)\( p^{29} T^{3} + p^{58} T^{4} \)
73$D_{4}$ \( 1 - \)\(99\!\cdots\!80\)\( T + \)\(15\!\cdots\!90\)\( T^{2} - \)\(99\!\cdots\!80\)\( p^{29} T^{3} + p^{58} T^{4} \)
79$D_{4}$ \( 1 + \)\(72\!\cdots\!80\)\( T + \)\(34\!\cdots\!38\)\( T^{2} + \)\(72\!\cdots\!80\)\( p^{29} T^{3} + p^{58} T^{4} \)
83$D_{4}$ \( 1 - \)\(22\!\cdots\!40\)\( T + \)\(60\!\cdots\!70\)\( T^{2} - \)\(22\!\cdots\!40\)\( p^{29} T^{3} + p^{58} T^{4} \)
89$D_{4}$ \( 1 - \)\(58\!\cdots\!60\)\( T + \)\(67\!\cdots\!18\)\( T^{2} - \)\(58\!\cdots\!60\)\( p^{29} T^{3} + p^{58} T^{4} \)
97$D_{4}$ \( 1 - \)\(10\!\cdots\!80\)\( T + \)\(96\!\cdots\!30\)\( T^{2} - \)\(10\!\cdots\!80\)\( p^{29} T^{3} + p^{58} T^{4} \)
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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

−25.72868657660485147030428912722, −23.85023141341665308092367887910, −22.99661949629539216345037865398, −22.91132095957644566528577857099, −21.58323327988740077110649105200, −19.93377438460368859215936465405, −18.82169509011267587663733755437, −17.98606264450376356129990123687, −15.92917042077896086436932580822, −15.79572792346414260372975867814, −13.38692374908591032596541219977, −12.86039942878683504422430358663, −11.22506451962109841988748838793, −9.637249782268918632286389013119, −7.949866560858724884326906739463, −6.12312213314869983040118889323, −4.46985241121272394578930720162, −3.34698244840578429721006084056, 0, 0, 3.34698244840578429721006084056, 4.46985241121272394578930720162, 6.12312213314869983040118889323, 7.949866560858724884326906739463, 9.637249782268918632286389013119, 11.22506451962109841988748838793, 12.86039942878683504422430358663, 13.38692374908591032596541219977, 15.79572792346414260372975867814, 15.92917042077896086436932580822, 17.98606264450376356129990123687, 18.82169509011267587663733755437, 19.93377438460368859215936465405, 21.58323327988740077110649105200, 22.91132095957644566528577857099, 22.99661949629539216345037865398, 23.85023141341665308092367887910, 25.72868657660485147030428912722

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