Properties

Label 4-3e2-1.1-c31e2-0-0
Degree $4$
Conductor $9$
Sign $1$
Analytic cond. $333.542$
Root an. cond. $4.27353$
Motivic weight $31$
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  − 3.95e4·2-s + 2.86e7·3-s − 1.46e9·4-s − 7.93e9·5-s − 1.13e12·6-s − 1.04e13·7-s + 9.28e13·8-s + 6.17e14·9-s + 3.13e14·10-s + 1.07e16·11-s − 4.20e16·12-s − 9.16e16·13-s + 4.14e17·14-s − 2.27e17·15-s − 4.82e17·16-s − 1.65e19·17-s − 2.44e19·18-s − 8.24e19·19-s + 1.16e19·20-s − 3.01e20·21-s − 4.26e20·22-s − 3.51e21·23-s + 2.66e21·24-s − 6.96e21·25-s + 3.62e21·26-s + 1.18e22·27-s + 1.53e22·28-s + ⋯
L(s)  = 1  − 0.852·2-s + 1.15·3-s − 0.683·4-s − 0.116·5-s − 0.984·6-s − 0.835·7-s + 0.932·8-s + 9-s + 0.0991·10-s + 0.778·11-s − 0.788·12-s − 0.496·13-s + 0.712·14-s − 0.134·15-s − 0.104·16-s − 1.40·17-s − 0.852·18-s − 1.24·19-s + 0.0793·20-s − 0.964·21-s − 0.663·22-s − 2.74·23-s + 1.07·24-s − 1.49·25-s + 0.423·26-s + 0.769·27-s + 0.570·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+31/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(333.542\)
Root analytic conductor: \(4.27353\)
Motivic weight: \(31\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 9,\ (\ :31/2, 31/2),\ 1)\)

Particular Values

\(L(16)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{33}{2})\) 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$C_1$ \( ( 1 - p^{15} T )^{2} \)
good2$D_{4}$ \( 1 + 4941 p^{3} T + 2958397 p^{10} T^{2} + 4941 p^{34} T^{3} + p^{62} T^{4} \)
5$D_{4}$ \( 1 + 1586103444 p T + 2247420944432827814 p^{5} T^{2} + 1586103444 p^{32} T^{3} + p^{62} T^{4} \)
7$D_{4}$ \( 1 + 1498410605168 p T + \)\(12\!\cdots\!46\)\( p^{4} T^{2} + 1498410605168 p^{32} T^{3} + p^{62} T^{4} \)
11$D_{4}$ \( 1 - 980187318266328 p T + \)\(88\!\cdots\!34\)\( p^{2} T^{2} - 980187318266328 p^{32} T^{3} + p^{62} T^{4} \)
13$D_{4}$ \( 1 + 7052829002619716 p T + \)\(26\!\cdots\!78\)\( p^{3} T^{2} + 7052829002619716 p^{32} T^{3} + p^{62} T^{4} \)
17$D_{4}$ \( 1 + 16595285055794710812 T + \)\(17\!\cdots\!06\)\( p T^{2} + 16595285055794710812 p^{31} T^{3} + p^{62} T^{4} \)
19$D_{4}$ \( 1 + 82480245798578318024 T + \)\(18\!\cdots\!38\)\( p^{2} T^{2} + 82480245798578318024 p^{31} T^{3} + p^{62} T^{4} \)
23$D_{4}$ \( 1 + \)\(35\!\cdots\!12\)\( T + \)\(26\!\cdots\!58\)\( p T^{2} + \)\(35\!\cdots\!12\)\( p^{31} T^{3} + p^{62} T^{4} \)
29$D_{4}$ \( 1 - \)\(94\!\cdots\!28\)\( p T + \)\(43\!\cdots\!38\)\( p^{2} T^{2} - \)\(94\!\cdots\!28\)\( p^{32} T^{3} + p^{62} T^{4} \)
31$D_{4}$ \( 1 - \)\(32\!\cdots\!92\)\( T - \)\(15\!\cdots\!38\)\( T^{2} - \)\(32\!\cdots\!92\)\( p^{31} T^{3} + p^{62} T^{4} \)
37$D_{4}$ \( 1 + \)\(35\!\cdots\!16\)\( T + \)\(84\!\cdots\!46\)\( T^{2} + \)\(35\!\cdots\!16\)\( p^{31} T^{3} + p^{62} T^{4} \)
41$D_{4}$ \( 1 + \)\(37\!\cdots\!08\)\( T + \)\(20\!\cdots\!82\)\( T^{2} + \)\(37\!\cdots\!08\)\( p^{31} T^{3} + p^{62} T^{4} \)
43$D_{4}$ \( 1 - \)\(26\!\cdots\!60\)\( T + \)\(10\!\cdots\!90\)\( T^{2} - \)\(26\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \)
47$D_{4}$ \( 1 - \)\(35\!\cdots\!20\)\( T + \)\(12\!\cdots\!90\)\( T^{2} - \)\(35\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \)
53$D_{4}$ \( 1 - \)\(20\!\cdots\!08\)\( T + \)\(33\!\cdots\!34\)\( T^{2} - \)\(20\!\cdots\!08\)\( p^{31} T^{3} + p^{62} T^{4} \)
59$D_{4}$ \( 1 - \)\(83\!\cdots\!44\)\( T + \)\(31\!\cdots\!98\)\( T^{2} - \)\(83\!\cdots\!44\)\( p^{31} T^{3} + p^{62} T^{4} \)
61$D_{4}$ \( 1 - \)\(71\!\cdots\!40\)\( p T + \)\(48\!\cdots\!38\)\( T^{2} - \)\(71\!\cdots\!40\)\( p^{32} T^{3} + p^{62} T^{4} \)
67$D_{4}$ \( 1 + \)\(42\!\cdots\!72\)\( T + \)\(10\!\cdots\!62\)\( T^{2} + \)\(42\!\cdots\!72\)\( p^{31} T^{3} + p^{62} T^{4} \)
71$D_{4}$ \( 1 + \)\(16\!\cdots\!96\)\( T + \)\(29\!\cdots\!46\)\( T^{2} + \)\(16\!\cdots\!96\)\( p^{31} T^{3} + p^{62} T^{4} \)
73$D_{4}$ \( 1 + \)\(11\!\cdots\!52\)\( T + \)\(12\!\cdots\!34\)\( T^{2} + \)\(11\!\cdots\!52\)\( p^{31} T^{3} + p^{62} T^{4} \)
79$D_{4}$ \( 1 + \)\(11\!\cdots\!00\)\( T + \)\(44\!\cdots\!58\)\( T^{2} + \)\(11\!\cdots\!00\)\( p^{31} T^{3} + p^{62} T^{4} \)
83$D_{4}$ \( 1 + \)\(12\!\cdots\!96\)\( T + \)\(97\!\cdots\!82\)\( T^{2} + \)\(12\!\cdots\!96\)\( p^{31} T^{3} + p^{62} T^{4} \)
89$D_{4}$ \( 1 + \)\(46\!\cdots\!04\)\( T + \)\(10\!\cdots\!98\)\( T^{2} + \)\(46\!\cdots\!04\)\( p^{31} T^{3} + p^{62} T^{4} \)
97$D_{4}$ \( 1 - \)\(96\!\cdots\!88\)\( T + \)\(97\!\cdots\!42\)\( T^{2} - \)\(96\!\cdots\!88\)\( p^{31} T^{3} + p^{62} T^{4} \)
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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

−17.68551850013125804367001956647, −17.67561900284002710512298734955, −16.15767680739050468337887796509, −15.49657468927713739582340493583, −14.31199601941628522589489653837, −13.73371302310480820357687378119, −12.90048748399637768454386053105, −11.82704705012575726832178691228, −10.03845988698785089401898209793, −9.842641148184267770987201065673, −8.605956664172178736323373405562, −8.528970930588172073533714884844, −7.18009746568930714560449307487, −6.13042128831849306310337388082, −4.20995343350361975982128564642, −3.99255245317134865122102547691, −2.48482734638277725353405184833, −1.68755407214605605303095610546, 0, 0, 1.68755407214605605303095610546, 2.48482734638277725353405184833, 3.99255245317134865122102547691, 4.20995343350361975982128564642, 6.13042128831849306310337388082, 7.18009746568930714560449307487, 8.528970930588172073533714884844, 8.605956664172178736323373405562, 9.842641148184267770987201065673, 10.03845988698785089401898209793, 11.82704705012575726832178691228, 12.90048748399637768454386053105, 13.73371302310480820357687378119, 14.31199601941628522589489653837, 15.49657468927713739582340493583, 16.15767680739050468337887796509, 17.67561900284002710512298734955, 17.68551850013125804367001956647

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