Properties

Label 4-3e2-1.1-c12e2-0-0
Degree $4$
Conductor $9$
Sign $1$
Analytic cond. $7.51846$
Root an. cond. $1.65589$
Motivic weight $12$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.35e3·3-s − 232·4-s + 8.05e4·7-s + 1.29e6·9-s + 3.13e5·12-s + 2.56e6·13-s − 1.67e7·16-s + 1.06e8·19-s − 1.08e8·21-s + 4.26e7·25-s − 1.02e9·27-s − 1.86e7·28-s + 1.33e8·31-s − 2.99e8·36-s + 4.45e9·37-s − 3.46e9·39-s + 1.79e10·43-s + 2.25e10·48-s − 2.28e10·49-s − 5.95e8·52-s − 1.44e11·57-s − 8.13e10·61-s + 1.03e11·63-s + 7.77e9·64-s + 2.42e11·67-s − 1.21e11·73-s − 5.75e10·75-s + ⋯
L(s)  = 1  − 1.85·3-s − 0.0566·4-s + 0.684·7-s + 2.42·9-s + 0.104·12-s + 0.532·13-s − 0.996·16-s + 2.26·19-s − 1.26·21-s + 0.174·25-s − 2.64·27-s − 0.0387·28-s + 0.149·31-s − 0.137·36-s + 1.73·37-s − 0.985·39-s + 2.84·43-s + 1.84·48-s − 1.64·49-s − 0.0301·52-s − 4.19·57-s − 1.57·61-s + 1.66·63-s + 0.113·64-s + 2.67·67-s − 0.805·73-s − 0.323·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+6)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(7.51846\)
Root analytic conductor: \(1.65589\)
Motivic weight: \(12\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9,\ (\ :6, 6),\ 1)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.9570928492\)
\(L(\frac12)\) \(\approx\) \(0.9570928492\)
\(L(7)\) 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$C_2$ \( 1 + 50 p^{3} T + p^{12} T^{2} \)
good2$C_2^2$ \( 1 + 29 p^{3} T^{2} + p^{24} T^{4} \)
5$C_2^2$ \( 1 - 1706066 p^{2} T^{2} + p^{24} T^{4} \)
7$C_2$ \( ( 1 - 5750 p T + p^{12} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 40734109202 p^{2} T^{2} + p^{24} T^{4} \)
13$C_2$ \( ( 1 - 1284050 T + p^{12} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 944884986604418 T^{2} + p^{24} T^{4} \)
19$C_2$ \( ( 1 - 53343578 T + p^{12} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 61021177942562 p^{2} T^{2} + p^{24} T^{4} \)
29$C_2^2$ \( 1 - 693172036445878082 T^{2} + p^{24} T^{4} \)
31$C_2$ \( ( 1 - 66526202 T + p^{12} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 60235850 p T + p^{12} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 22304779456187067838 T^{2} + p^{24} T^{4} \)
43$C_2$ \( ( 1 - 8977216250 T + p^{12} T^{2} )^{2} \)
47$C_2^2$ \( 1 - \)\(23\!\cdots\!78\)\( T^{2} + p^{24} T^{4} \)
53$C_2^2$ \( 1 + \)\(71\!\cdots\!22\)\( T^{2} + p^{24} T^{4} \)
59$C_2^2$ \( 1 - \)\(14\!\cdots\!62\)\( T^{2} + p^{24} T^{4} \)
61$C_2$ \( ( 1 + 40679935918 T + p^{12} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 121176846650 T + p^{12} T^{2} )^{2} \)
71$C_2^2$ \( 1 - \)\(30\!\cdots\!82\)\( T^{2} + p^{24} T^{4} \)
73$C_2$ \( ( 1 + 60956187550 T + p^{12} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 252324997702 T + p^{12} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(45\!\cdots\!38\)\( T^{2} + p^{24} T^{4} \)
89$C_2^2$ \( 1 - \)\(48\!\cdots\!42\)\( T^{2} + p^{24} T^{4} \)
97$C_2$ \( ( 1 - 653817778850 T + p^{12} T^{2} )^{2} \)
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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

−24.46060255202734799575596320719, −23.28240514060699706949461507467, −22.69971702967702057995006089881, −22.07097712265710489133444388825, −21.12006835019698744233446763829, −20.24456941976864783445144189082, −18.61987516488622206935257128072, −18.03274477295533426490251460125, −17.43035593622051377478465341535, −16.23295800461152333481842013949, −15.75789095111368458466305609757, −14.06103303664286913266335788310, −12.77697009445635526237317130872, −11.54389191159432437564054235340, −11.10897317795989106351748506705, −9.618781225369489204210340327025, −7.42240927695159248789973113821, −5.91740399460202559386893852271, −4.66217614320510640493547412182, −0.981697946157363364975853116385, 0.981697946157363364975853116385, 4.66217614320510640493547412182, 5.91740399460202559386893852271, 7.42240927695159248789973113821, 9.618781225369489204210340327025, 11.10897317795989106351748506705, 11.54389191159432437564054235340, 12.77697009445635526237317130872, 14.06103303664286913266335788310, 15.75789095111368458466305609757, 16.23295800461152333481842013949, 17.43035593622051377478465341535, 18.03274477295533426490251460125, 18.61987516488622206935257128072, 20.24456941976864783445144189082, 21.12006835019698744233446763829, 22.07097712265710489133444388825, 22.69971702967702057995006089881, 23.28240514060699706949461507467, 24.46060255202734799575596320719

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