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Properties

Label 4-3e2-1.1-c25e2-0-0
Degree $4$
Conductor $9$
Sign $1$
Analytic cond. $141.132$
Root an. cond. $3.44672$
Motivic weight $25$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 324·2-s − 1.06e6·3-s − 2.06e7·4-s + 5.70e8·5-s + 3.44e8·6-s − 2.96e10·7-s + 2.57e9·8-s + 8.47e11·9-s − 1.84e11·10-s − 1.81e13·11-s + 2.19e13·12-s − 1.05e14·13-s + 9.61e12·14-s − 6.06e14·15-s − 6.92e14·16-s − 4.56e14·17-s − 2.74e14·18-s + 1.38e16·19-s − 1.18e16·20-s + 3.15e16·21-s + 5.89e15·22-s + 1.39e17·23-s − 2.73e15·24-s − 3.27e17·25-s + 3.40e16·26-s − 6.00e17·27-s + 6.14e17·28-s + ⋯
L(s)  = 1  − 0.0559·2-s − 1.15·3-s − 0.616·4-s + 1.04·5-s + 0.0645·6-s − 0.810·7-s + 0.0132·8-s + 9-s − 0.0584·10-s − 1.74·11-s + 0.712·12-s − 1.25·13-s + 0.0453·14-s − 1.20·15-s − 0.615·16-s − 0.190·17-s − 0.0559·18-s + 1.44·19-s − 0.645·20-s + 0.936·21-s + 0.0977·22-s + 1.32·23-s − 0.0152·24-s − 1.09·25-s + 0.0700·26-s − 0.769·27-s + 0.500·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(141.132\)
Root analytic conductor: \(3.44672\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 9,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{27}{2})\) 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$C_1$ \( ( 1 + p^{12} T )^{2} \)
good2$D_{4}$ \( 1 + 81 p^{2} T + 40631 p^{9} T^{2} + 81 p^{27} T^{3} + p^{50} T^{4} \)
5$D_{4}$ \( 1 - 570861756 T + 26127791204487934 p^{2} T^{2} - 570861756 p^{25} T^{3} + p^{50} T^{4} \)
7$D_{4}$ \( 1 + 4241055104 p T + 696480056120365134 p^{4} T^{2} + 4241055104 p^{26} T^{3} + p^{50} T^{4} \)
11$D_{4}$ \( 1 + 1653435056232 p T + \)\(19\!\cdots\!14\)\( p^{3} T^{2} + 1653435056232 p^{26} T^{3} + p^{50} T^{4} \)
13$D_{4}$ \( 1 + 105143636679236 T + \)\(10\!\cdots\!98\)\( p T^{2} + 105143636679236 p^{25} T^{3} + p^{50} T^{4} \)
17$D_{4}$ \( 1 + 456987364349436 T + \)\(41\!\cdots\!14\)\( p T^{2} + 456987364349436 p^{25} T^{3} + p^{50} T^{4} \)
19$D_{4}$ \( 1 - 731444131334056 p T + \)\(24\!\cdots\!98\)\( p^{2} T^{2} - 731444131334056 p^{26} T^{3} + p^{50} T^{4} \)
23$D_{4}$ \( 1 - 6058360821754512 p T + \)\(15\!\cdots\!74\)\( p^{2} T^{2} - 6058360821754512 p^{26} T^{3} + p^{50} T^{4} \)
29$D_{4}$ \( 1 + 2502033490260709812 T + \)\(80\!\cdots\!98\)\( T^{2} + 2502033490260709812 p^{25} T^{3} + p^{50} T^{4} \)
31$D_{4}$ \( 1 - 1075472102492455312 T - \)\(12\!\cdots\!98\)\( T^{2} - 1075472102492455312 p^{25} T^{3} + p^{50} T^{4} \)
37$D_{4}$ \( 1 + 1971470949238312628 T + \)\(32\!\cdots\!94\)\( T^{2} + 1971470949238312628 p^{25} T^{3} + p^{50} T^{4} \)
41$D_{4}$ \( 1 + \)\(41\!\cdots\!68\)\( T + \)\(84\!\cdots\!62\)\( T^{2} + \)\(41\!\cdots\!68\)\( p^{25} T^{3} + p^{50} T^{4} \)
43$D_{4}$ \( 1 - 38930395898259827080 T + \)\(43\!\cdots\!70\)\( T^{2} - 38930395898259827080 p^{25} T^{3} + p^{50} T^{4} \)
47$D_{4}$ \( 1 + \)\(73\!\cdots\!40\)\( T + \)\(14\!\cdots\!10\)\( T^{2} + \)\(73\!\cdots\!40\)\( p^{25} T^{3} + p^{50} T^{4} \)
53$D_{4}$ \( 1 + \)\(84\!\cdots\!64\)\( T + \)\(43\!\cdots\!86\)\( T^{2} + \)\(84\!\cdots\!64\)\( p^{25} T^{3} + p^{50} T^{4} \)
59$D_{4}$ \( 1 - \)\(84\!\cdots\!96\)\( T + \)\(38\!\cdots\!78\)\( T^{2} - \)\(84\!\cdots\!96\)\( p^{25} T^{3} + p^{50} T^{4} \)
61$D_{4}$ \( 1 + \)\(24\!\cdots\!60\)\( T + \)\(72\!\cdots\!58\)\( T^{2} + \)\(24\!\cdots\!60\)\( p^{25} T^{3} + p^{50} T^{4} \)
67$D_{4}$ \( 1 + \)\(13\!\cdots\!56\)\( T + \)\(10\!\cdots\!98\)\( T^{2} + \)\(13\!\cdots\!56\)\( p^{25} T^{3} + p^{50} T^{4} \)
71$D_{4}$ \( 1 - \)\(12\!\cdots\!84\)\( T + \)\(25\!\cdots\!66\)\( T^{2} - \)\(12\!\cdots\!84\)\( p^{25} T^{3} + p^{50} T^{4} \)
73$D_{4}$ \( 1 - \)\(84\!\cdots\!76\)\( T + \)\(70\!\cdots\!26\)\( T^{2} - \)\(84\!\cdots\!76\)\( p^{25} T^{3} + p^{50} T^{4} \)
79$D_{4}$ \( 1 - \)\(12\!\cdots\!80\)\( T + \)\(91\!\cdots\!98\)\( T^{2} - \)\(12\!\cdots\!80\)\( p^{25} T^{3} + p^{50} T^{4} \)
83$D_{4}$ \( 1 + \)\(87\!\cdots\!12\)\( T + \)\(20\!\cdots\!58\)\( T^{2} + \)\(87\!\cdots\!12\)\( p^{25} T^{3} + p^{50} T^{4} \)
89$D_{4}$ \( 1 - \)\(21\!\cdots\!44\)\( T + \)\(11\!\cdots\!18\)\( T^{2} - \)\(21\!\cdots\!44\)\( p^{25} T^{3} + p^{50} T^{4} \)
97$D_{4}$ \( 1 + \)\(55\!\cdots\!16\)\( T + \)\(10\!\cdots\!78\)\( T^{2} + \)\(55\!\cdots\!16\)\( p^{25} T^{3} + p^{50} T^{4} \)
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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

−19.09575194001369631085066621019, −18.31800289609088332910017940263, −17.72926106219290764723810215068, −17.01873994671008972642925396430, −16.09161489194395351973017042675, −15.21363142276009188531192342459, −13.47283324448360023906937134211, −13.32700988320387336647781878333, −12.18758262511362255973745946056, −10.97677971301561767639065507078, −9.785078884342510845822047664472, −9.590651031119803877311337379530, −7.65801652679983552938765927446, −6.54383119547105954998142968596, −5.28494678642492329572697611963, −5.01374408911311389881791562545, −3.07143586041900644537713510962, −1.78130573606761996021613258231, 0, 0, 1.78130573606761996021613258231, 3.07143586041900644537713510962, 5.01374408911311389881791562545, 5.28494678642492329572697611963, 6.54383119547105954998142968596, 7.65801652679983552938765927446, 9.590651031119803877311337379530, 9.785078884342510845822047664472, 10.97677971301561767639065507078, 12.18758262511362255973745946056, 13.32700988320387336647781878333, 13.47283324448360023906937134211, 15.21363142276009188531192342459, 16.09161489194395351973017042675, 17.01873994671008972642925396430, 17.72926106219290764723810215068, 18.31800289609088332910017940263, 19.09575194001369631085066621019

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