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Properties

Label 4-1-1.1-c27e2-0-0
Degree $4$
Conductor $1$
Sign $1$
Analytic cond. $21.3310$
Root an. cond. $2.14908$
Motivic weight $27$
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  − 8.28e3·2-s − 1.28e6·3-s − 4.62e6·4-s + 5.44e9·5-s + 1.06e10·6-s − 1.75e11·7-s − 4.67e11·8-s − 6.18e12·9-s − 4.50e13·10-s + 1.38e14·11-s + 5.95e12·12-s − 7.53e14·13-s + 1.45e15·14-s − 7.00e15·15-s − 4.60e15·16-s − 2.97e16·17-s + 5.11e16·18-s + 4.04e17·19-s − 2.51e16·20-s + 2.25e17·21-s − 1.14e18·22-s + 2.92e18·23-s + 6.00e17·24-s + 1.19e19·25-s + 6.23e18·26-s + 8.21e18·27-s + 8.11e17·28-s + ⋯
L(s)  = 1  − 0.714·2-s − 0.465·3-s − 0.0344·4-s + 1.99·5-s + 0.332·6-s − 0.684·7-s − 0.300·8-s − 0.810·9-s − 1.42·10-s + 1.20·11-s + 0.0160·12-s − 0.689·13-s + 0.489·14-s − 0.928·15-s − 0.255·16-s − 0.728·17-s + 0.579·18-s + 2.20·19-s − 0.0687·20-s + 0.318·21-s − 0.862·22-s + 1.21·23-s + 0.139·24-s + 1.60·25-s + 0.493·26-s + 0.390·27-s + 0.0235·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(28-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+27/2)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(21.3310\)
Root analytic conductor: \(2.14908\)
Motivic weight: \(27\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1,\ (\ :27/2, 27/2),\ 1)\)

Particular Values

\(L(14)\) \(\approx\) \(1.239199108\)
\(L(\frac12)\) \(\approx\) \(1.239199108\)
\(L(\frac{29}{2})\) 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$D_{4}$ \( 1 + 1035 p^{3} T + 35735 p^{11} T^{2} + 1035 p^{30} T^{3} + p^{54} T^{4} \)
3$D_{4}$ \( 1 + 15880 p^{4} T + 1194221510 p^{8} T^{2} + 15880 p^{31} T^{3} + p^{54} T^{4} \)
5$D_{4}$ \( 1 - 217743516 p^{2} T + 5666334661152134 p^{5} T^{2} - 217743516 p^{29} T^{3} + p^{54} T^{4} \)
7$D_{4}$ \( 1 + 25055994800 p T + 52976088888041064750 p^{4} T^{2} + 25055994800 p^{28} T^{3} + p^{54} T^{4} \)
11$D_{4}$ \( 1 - 12560667062904 p T + \)\(90\!\cdots\!46\)\( p^{3} T^{2} - 12560667062904 p^{28} T^{3} + p^{54} T^{4} \)
13$D_{4}$ \( 1 + 57956446251620 p T + \)\(14\!\cdots\!30\)\( p^{2} T^{2} + 57956446251620 p^{28} T^{3} + p^{54} T^{4} \)
17$D_{4}$ \( 1 + 29753620331011740 T + \)\(20\!\cdots\!70\)\( p T^{2} + 29753620331011740 p^{27} T^{3} + p^{54} T^{4} \)
19$D_{4}$ \( 1 - 21292937388036040 p T + \)\(29\!\cdots\!98\)\( p^{2} T^{2} - 21292937388036040 p^{28} T^{3} + p^{54} T^{4} \)
23$D_{4}$ \( 1 - 127351257527009520 p T + \)\(22\!\cdots\!70\)\( p^{2} T^{2} - 127351257527009520 p^{28} T^{3} + p^{54} T^{4} \)
29$D_{4}$ \( 1 + 15546679995448558260 T + \)\(61\!\cdots\!18\)\( T^{2} + 15546679995448558260 p^{27} T^{3} + p^{54} T^{4} \)
31$D_{4}$ \( 1 - 28544554594467385024 T + \)\(31\!\cdots\!66\)\( T^{2} - 28544554594467385024 p^{27} T^{3} + p^{54} T^{4} \)
37$D_{4}$ \( 1 - \)\(18\!\cdots\!80\)\( T + \)\(49\!\cdots\!70\)\( T^{2} - \)\(18\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
41$D_{4}$ \( 1 - \)\(90\!\cdots\!64\)\( T + \)\(89\!\cdots\!86\)\( T^{2} - \)\(90\!\cdots\!64\)\( p^{27} T^{3} + p^{54} T^{4} \)
43$D_{4}$ \( 1 - \)\(51\!\cdots\!00\)\( T + \)\(17\!\cdots\!50\)\( T^{2} - \)\(51\!\cdots\!00\)\( p^{27} T^{3} + p^{54} T^{4} \)
47$D_{4}$ \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(27\!\cdots\!10\)\( T^{2} + \)\(11\!\cdots\!60\)\( p^{27} T^{3} + p^{54} T^{4} \)
53$D_{4}$ \( 1 - \)\(10\!\cdots\!20\)\( T + \)\(63\!\cdots\!10\)\( T^{2} - \)\(10\!\cdots\!20\)\( p^{27} T^{3} + p^{54} T^{4} \)
59$D_{4}$ \( 1 - \)\(20\!\cdots\!80\)\( T + \)\(23\!\cdots\!38\)\( T^{2} - \)\(20\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
61$D_{4}$ \( 1 - \)\(14\!\cdots\!44\)\( T + \)\(12\!\cdots\!26\)\( T^{2} - \)\(14\!\cdots\!44\)\( p^{27} T^{3} + p^{54} T^{4} \)
67$D_{4}$ \( 1 - \)\(30\!\cdots\!60\)\( T + \)\(20\!\cdots\!90\)\( T^{2} - \)\(30\!\cdots\!60\)\( p^{27} T^{3} + p^{54} T^{4} \)
71$D_{4}$ \( 1 + \)\(13\!\cdots\!16\)\( T + \)\(13\!\cdots\!46\)\( T^{2} + \)\(13\!\cdots\!16\)\( p^{27} T^{3} + p^{54} T^{4} \)
73$D_{4}$ \( 1 - \)\(52\!\cdots\!60\)\( T + \)\(30\!\cdots\!30\)\( T^{2} - \)\(52\!\cdots\!60\)\( p^{27} T^{3} + p^{54} T^{4} \)
79$D_{4}$ \( 1 - \)\(62\!\cdots\!40\)\( T + \)\(41\!\cdots\!18\)\( T^{2} - \)\(62\!\cdots\!40\)\( p^{27} T^{3} + p^{54} T^{4} \)
83$D_{4}$ \( 1 - \)\(17\!\cdots\!80\)\( T + \)\(20\!\cdots\!90\)\( T^{2} - \)\(17\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
89$D_{4}$ \( 1 + \)\(31\!\cdots\!80\)\( T + \)\(11\!\cdots\!58\)\( T^{2} + \)\(31\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
97$D_{4}$ \( 1 + \)\(65\!\cdots\!60\)\( T + \)\(48\!\cdots\!10\)\( T^{2} + \)\(65\!\cdots\!60\)\( p^{27} T^{3} + p^{54} T^{4} \)
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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

−27.10563959812033108832731117675, −26.24073429598295890148619961947, −25.11750685583981067778108451479, −24.64414644958356609087659196152, −22.50188545029103663195437511378, −22.21892371120360750818924346610, −20.79737968853638185831337102591, −19.50265774061765824385221797449, −17.84624530116015061782129782870, −17.59210966024527609444600839401, −16.50582946123084970950610092127, −14.39270451750690737940576636597, −13.30841394829815483779194433379, −11.52175848608226414749693302062, −9.537244640310907359419777634061, −9.348954137281175526758391198685, −6.56510505470169432955925772781, −5.49504866683922240527961455874, −2.60840721330549053446694296621, −0.877191859570262290319324466835, 0.877191859570262290319324466835, 2.60840721330549053446694296621, 5.49504866683922240527961455874, 6.56510505470169432955925772781, 9.348954137281175526758391198685, 9.537244640310907359419777634061, 11.52175848608226414749693302062, 13.30841394829815483779194433379, 14.39270451750690737940576636597, 16.50582946123084970950610092127, 17.59210966024527609444600839401, 17.84624530116015061782129782870, 19.50265774061765824385221797449, 20.79737968853638185831337102591, 22.21892371120360750818924346610, 22.50188545029103663195437511378, 24.64414644958356609087659196152, 25.11750685583981067778108451479, 26.24073429598295890148619961947, 27.10563959812033108832731117675

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