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Properties

Label 4-3e2-1.1-c29e2-0-0
Degree $4$
Conductor $9$
Sign $1$
Analytic cond. $255.469$
Root an. cond. $3.99792$
Motivic weight $29$
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  − 4.50e4·2-s − 9.56e6·3-s + 5.83e8·4-s + 1.87e8·5-s + 4.30e11·6-s + 2.64e12·7-s + 1.46e13·8-s + 6.86e13·9-s − 8.44e12·10-s + 1.18e15·11-s − 5.58e15·12-s − 2.16e16·13-s − 1.19e17·14-s − 1.79e15·15-s − 7.00e17·16-s − 3.03e17·17-s − 3.09e18·18-s − 7.82e18·19-s + 1.09e17·20-s − 2.52e19·21-s − 5.32e19·22-s − 1.11e18·23-s − 1.39e20·24-s + 1.07e20·25-s + 9.73e20·26-s − 4.37e20·27-s + 1.54e21·28-s + ⋯
L(s)  = 1  − 1.94·2-s − 1.15·3-s + 1.08·4-s + 0.0137·5-s + 2.24·6-s + 1.47·7-s + 1.17·8-s + 9-s − 0.0267·10-s + 0.939·11-s − 1.25·12-s − 1.52·13-s − 2.86·14-s − 0.0158·15-s − 2.43·16-s − 0.437·17-s − 1.94·18-s − 2.24·19-s + 0.0149·20-s − 1.70·21-s − 1.82·22-s − 0.0200·23-s − 1.35·24-s + 0.576·25-s + 2.96·26-s − 0.769·27-s + 1.60·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+29/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(255.469\)
Root analytic conductor: \(3.99792\)
Motivic weight: \(29\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 9,\ (\ :29/2, 29/2),\ 1)\)

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)$
bad3$C_1$ \( ( 1 + p^{14} T )^{2} \)
good2$D_{4}$ \( 1 + 11259 p^{2} T + 5643817 p^{8} T^{2} + 11259 p^{31} T^{3} + p^{58} T^{4} \)
5$D_{4}$ \( 1 - 37522764 p T - 171762472153584242 p^{4} T^{2} - 37522764 p^{30} T^{3} + p^{58} T^{4} \)
7$D_{4}$ \( 1 - 53958696352 p^{2} T + \)\(17\!\cdots\!74\)\( p^{4} T^{2} - 53958696352 p^{31} T^{3} + p^{58} T^{4} \)
11$D_{4}$ \( 1 - 1183469847829128 T + \)\(27\!\cdots\!14\)\( p T^{2} - 1183469847829128 p^{29} T^{3} + p^{58} T^{4} \)
13$D_{4}$ \( 1 + 1663145265242228 p T + \)\(15\!\cdots\!82\)\( p^{3} T^{2} + 1663145265242228 p^{30} T^{3} + p^{58} T^{4} \)
17$D_{4}$ \( 1 + 303436852359947964 T + \)\(72\!\cdots\!18\)\( T^{2} + 303436852359947964 p^{29} T^{3} + p^{58} T^{4} \)
19$D_{4}$ \( 1 + 411994174117924184 p T + \)\(56\!\cdots\!82\)\( p^{3} T^{2} + 411994174117924184 p^{30} T^{3} + p^{58} T^{4} \)
23$D_{4}$ \( 1 + 1116121153769543376 T + \)\(25\!\cdots\!22\)\( p T^{2} + 1116121153769543376 p^{29} T^{3} + p^{58} T^{4} \)
29$D_{4}$ \( 1 + \)\(15\!\cdots\!92\)\( T + \)\(43\!\cdots\!18\)\( T^{2} + \)\(15\!\cdots\!92\)\( p^{29} T^{3} + p^{58} T^{4} \)
31$D_{4}$ \( 1 - \)\(51\!\cdots\!12\)\( T + \)\(40\!\cdots\!42\)\( T^{2} - \)\(51\!\cdots\!12\)\( p^{29} T^{3} + p^{58} T^{4} \)
37$D_{4}$ \( 1 + \)\(11\!\cdots\!12\)\( T + \)\(81\!\cdots\!34\)\( T^{2} + \)\(11\!\cdots\!12\)\( p^{29} T^{3} + p^{58} T^{4} \)
41$D_{4}$ \( 1 - \)\(32\!\cdots\!32\)\( T + \)\(13\!\cdots\!62\)\( T^{2} - \)\(32\!\cdots\!32\)\( p^{29} T^{3} + p^{58} T^{4} \)
43$D_{4}$ \( 1 + \)\(44\!\cdots\!00\)\( T + \)\(40\!\cdots\!10\)\( T^{2} + \)\(44\!\cdots\!00\)\( p^{29} T^{3} + p^{58} T^{4} \)
47$D_{4}$ \( 1 + \)\(34\!\cdots\!60\)\( T + \)\(86\!\cdots\!10\)\( T^{2} + \)\(34\!\cdots\!60\)\( p^{29} T^{3} + p^{58} T^{4} \)
53$D_{4}$ \( 1 + \)\(14\!\cdots\!96\)\( T + \)\(17\!\cdots\!86\)\( T^{2} + \)\(14\!\cdots\!96\)\( p^{29} T^{3} + p^{58} T^{4} \)
59$D_{4}$ \( 1 + \)\(82\!\cdots\!76\)\( p T + \)\(50\!\cdots\!18\)\( T^{2} + \)\(82\!\cdots\!76\)\( p^{30} T^{3} + p^{58} T^{4} \)
61$D_{4}$ \( 1 + \)\(74\!\cdots\!20\)\( T + \)\(13\!\cdots\!98\)\( T^{2} + \)\(74\!\cdots\!20\)\( p^{29} T^{3} + p^{58} T^{4} \)
67$D_{4}$ \( 1 + \)\(36\!\cdots\!64\)\( T + \)\(10\!\cdots\!18\)\( T^{2} + \)\(36\!\cdots\!64\)\( p^{29} T^{3} + p^{58} T^{4} \)
71$D_{4}$ \( 1 + \)\(12\!\cdots\!56\)\( T + \)\(71\!\cdots\!46\)\( T^{2} + \)\(12\!\cdots\!56\)\( p^{29} T^{3} + p^{58} T^{4} \)
73$D_{4}$ \( 1 + \)\(73\!\cdots\!76\)\( T + \)\(31\!\cdots\!62\)\( p T^{2} + \)\(73\!\cdots\!76\)\( p^{29} T^{3} + p^{58} T^{4} \)
79$D_{4}$ \( 1 + \)\(52\!\cdots\!40\)\( T + \)\(19\!\cdots\!38\)\( T^{2} + \)\(52\!\cdots\!40\)\( p^{29} T^{3} + p^{58} T^{4} \)
83$D_{4}$ \( 1 - \)\(79\!\cdots\!12\)\( T + \)\(10\!\cdots\!18\)\( T^{2} - \)\(79\!\cdots\!12\)\( p^{29} T^{3} + p^{58} T^{4} \)
89$D_{4}$ \( 1 - \)\(12\!\cdots\!24\)\( T + \)\(52\!\cdots\!18\)\( T^{2} - \)\(12\!\cdots\!24\)\( p^{29} T^{3} + p^{58} T^{4} \)
97$D_{4}$ \( 1 + \)\(92\!\cdots\!44\)\( T + \)\(77\!\cdots\!18\)\( T^{2} + \)\(92\!\cdots\!44\)\( p^{29} T^{3} + p^{58} T^{4} \)
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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

−18.02072631643023633171017466894, −17.54325833699976025160366699611, −17.10835813026695821590247734818, −16.69926870683345798316949056247, −15.11524088096884371121288344158, −14.24786298158883391104484561517, −12.76971412813786345004273270846, −11.69603860539765911565172034807, −10.83852301463986979301777009920, −10.21597040378714655350991290085, −9.179276186295923644281376754626, −8.397417845318416521745823377127, −7.51046506106755276279342566048, −6.49060567627457641714563629186, −4.78248232069624325728104945710, −4.52380123102539489709643759854, −1.84917962322803316114029015551, −1.39184614727213788151250460765, 0, 0, 1.39184614727213788151250460765, 1.84917962322803316114029015551, 4.52380123102539489709643759854, 4.78248232069624325728104945710, 6.49060567627457641714563629186, 7.51046506106755276279342566048, 8.397417845318416521745823377127, 9.179276186295923644281376754626, 10.21597040378714655350991290085, 10.83852301463986979301777009920, 11.69603860539765911565172034807, 12.76971412813786345004273270846, 14.24786298158883391104484561517, 15.11524088096884371121288344158, 16.69926870683345798316949056247, 17.10835813026695821590247734818, 17.54325833699976025160366699611, 18.02072631643023633171017466894

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