Properties

Degree $4$
Conductor $9$
Sign $1$
Motivic weight $35$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.09e4·2-s + 2.58e8·3-s − 3.93e9·4-s − 1.33e12·5-s − 1.57e13·6-s − 1.20e15·7-s − 1.38e15·8-s + 5.00e16·9-s + 8.12e16·10-s − 1.47e18·11-s − 1.01e18·12-s + 3.00e19·13-s + 7.31e19·14-s − 3.44e20·15-s − 9.38e20·16-s + 6.18e21·17-s − 3.04e21·18-s − 1.60e22·19-s + 5.24e21·20-s − 3.10e23·21-s + 8.98e22·22-s + 4.93e23·23-s − 3.58e23·24-s − 4.39e24·25-s − 1.82e24·26-s + 8.61e24·27-s + 4.72e24·28-s + ⋯
L(s)  = 1  − 0.328·2-s + 1.15·3-s − 0.114·4-s − 0.781·5-s − 0.379·6-s − 1.95·7-s − 0.217·8-s + 9-s + 0.256·10-s − 0.879·11-s − 0.132·12-s + 0.962·13-s + 0.641·14-s − 0.902·15-s − 0.794·16-s + 1.81·17-s − 0.328·18-s − 0.671·19-s + 0.0894·20-s − 2.25·21-s + 0.289·22-s + 0.730·23-s − 0.251·24-s − 1.51·25-s − 0.316·26-s + 0.769·27-s + 0.223·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+35/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Motivic weight: \(35\)
Character: induced by $\chi_{3} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 9,\ (\ :35/2, 35/2),\ 1)\)

Particular Values

\(L(18)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{37}{2})\) 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$C_1$ \( ( 1 - p^{17} T )^{2} \)
good2$D_{4}$ \( 1 + 3807 p^{4} T + 3732131 p^{11} T^{2} + 3807 p^{39} T^{3} + p^{70} T^{4} \)
5$D_{4}$ \( 1 + 266755899348 p T + \)\(19\!\cdots\!58\)\( p^{5} T^{2} + 266755899348 p^{36} T^{3} + p^{70} T^{4} \)
7$D_{4}$ \( 1 + 24520669039856 p^{2} T + \)\(46\!\cdots\!86\)\( p^{4} T^{2} + 24520669039856 p^{37} T^{3} + p^{70} T^{4} \)
11$D_{4}$ \( 1 + 1474443852221320632 T + \)\(44\!\cdots\!54\)\( p^{2} T^{2} + 1474443852221320632 p^{35} T^{3} + p^{70} T^{4} \)
13$D_{4}$ \( 1 - 2308093358323513756 p T + \)\(25\!\cdots\!98\)\( p^{3} T^{2} - 2308093358323513756 p^{36} T^{3} + p^{70} T^{4} \)
17$D_{4}$ \( 1 - \)\(36\!\cdots\!96\)\( p T + \)\(10\!\cdots\!78\)\( p^{2} T^{2} - \)\(36\!\cdots\!96\)\( p^{36} T^{3} + p^{70} T^{4} \)
19$D_{4}$ \( 1 + \)\(16\!\cdots\!64\)\( T + \)\(29\!\cdots\!62\)\( p T^{2} + \)\(16\!\cdots\!64\)\( p^{35} T^{3} + p^{70} T^{4} \)
23$D_{4}$ \( 1 - \)\(49\!\cdots\!72\)\( T + \)\(39\!\cdots\!14\)\( T^{2} - \)\(49\!\cdots\!72\)\( p^{35} T^{3} + p^{70} T^{4} \)
29$D_{4}$ \( 1 + \)\(78\!\cdots\!48\)\( T + \)\(15\!\cdots\!22\)\( p T^{2} + \)\(78\!\cdots\!48\)\( p^{35} T^{3} + p^{70} T^{4} \)
31$D_{4}$ \( 1 + \)\(12\!\cdots\!68\)\( p^{2} T + \)\(35\!\cdots\!82\)\( p^{2} T^{2} + \)\(12\!\cdots\!68\)\( p^{37} T^{3} + p^{70} T^{4} \)
37$D_{4}$ \( 1 - \)\(16\!\cdots\!16\)\( T + \)\(20\!\cdots\!06\)\( T^{2} - \)\(16\!\cdots\!16\)\( p^{35} T^{3} + p^{70} T^{4} \)
41$D_{4}$ \( 1 + \)\(32\!\cdots\!88\)\( T + \)\(68\!\cdots\!22\)\( T^{2} + \)\(32\!\cdots\!88\)\( p^{35} T^{3} + p^{70} T^{4} \)
43$D_{4}$ \( 1 - \)\(10\!\cdots\!20\)\( T - \)\(10\!\cdots\!10\)\( T^{2} - \)\(10\!\cdots\!20\)\( p^{35} T^{3} + p^{70} T^{4} \)
47$D_{4}$ \( 1 + \)\(52\!\cdots\!20\)\( T + \)\(66\!\cdots\!90\)\( T^{2} + \)\(52\!\cdots\!20\)\( p^{35} T^{3} + p^{70} T^{4} \)
53$D_{4}$ \( 1 + \)\(31\!\cdots\!68\)\( T - \)\(16\!\cdots\!26\)\( T^{2} + \)\(31\!\cdots\!68\)\( p^{35} T^{3} + p^{70} T^{4} \)
59$D_{4}$ \( 1 - \)\(82\!\cdots\!64\)\( T + \)\(20\!\cdots\!58\)\( T^{2} - \)\(82\!\cdots\!64\)\( p^{35} T^{3} + p^{70} T^{4} \)
61$D_{4}$ \( 1 + \)\(40\!\cdots\!40\)\( T + \)\(98\!\cdots\!18\)\( T^{2} + \)\(40\!\cdots\!40\)\( p^{35} T^{3} + p^{70} T^{4} \)
67$D_{4}$ \( 1 - \)\(96\!\cdots\!12\)\( T + \)\(11\!\cdots\!22\)\( T^{2} - \)\(96\!\cdots\!12\)\( p^{35} T^{3} + p^{70} T^{4} \)
71$D_{4}$ \( 1 - \)\(44\!\cdots\!24\)\( T + \)\(99\!\cdots\!46\)\( T^{2} - \)\(44\!\cdots\!24\)\( p^{35} T^{3} + p^{70} T^{4} \)
73$D_{4}$ \( 1 + \)\(13\!\cdots\!08\)\( T + \)\(29\!\cdots\!54\)\( T^{2} + \)\(13\!\cdots\!08\)\( p^{35} T^{3} + p^{70} T^{4} \)
79$D_{4}$ \( 1 + \)\(10\!\cdots\!80\)\( T + \)\(29\!\cdots\!98\)\( T^{2} + \)\(10\!\cdots\!80\)\( p^{35} T^{3} + p^{70} T^{4} \)
83$D_{4}$ \( 1 + \)\(55\!\cdots\!44\)\( T + \)\(36\!\cdots\!82\)\( T^{2} + \)\(55\!\cdots\!44\)\( p^{35} T^{3} + p^{70} T^{4} \)
89$D_{4}$ \( 1 - \)\(24\!\cdots\!36\)\( T + \)\(40\!\cdots\!58\)\( T^{2} - \)\(24\!\cdots\!36\)\( p^{35} T^{3} + p^{70} T^{4} \)
97$D_{4}$ \( 1 - \)\(83\!\cdots\!52\)\( T + \)\(66\!\cdots\!62\)\( T^{2} - \)\(83\!\cdots\!52\)\( p^{35} T^{3} + p^{70} T^{4} \)
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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

−16.81292288933172053374390668519, −16.04495064671967522951730817371, −15.54018395149442515258646346459, −14.74964043303869700742126825607, −13.31385302566793651356660744058, −13.21601293059975712348746058460, −12.13806351546674382447127171779, −10.86179711063681571934403597889, −9.694634232283678999756323381430, −9.384661906935522444193449251363, −8.274855955209209189422562454424, −7.60188097945322164421779557895, −6.62310711375376777026040628729, −5.55575620762707654186801387240, −3.85875687224132159374955368759, −3.48928380747504301354951356939, −2.75682651962078438686755235773, −1.56292338907903054527095736241, 0, 0, 1.56292338907903054527095736241, 2.75682651962078438686755235773, 3.48928380747504301354951356939, 3.85875687224132159374955368759, 5.55575620762707654186801387240, 6.62310711375376777026040628729, 7.60188097945322164421779557895, 8.274855955209209189422562454424, 9.384661906935522444193449251363, 9.694634232283678999756323381430, 10.86179711063681571934403597889, 12.13806351546674382447127171779, 13.21601293059975712348746058460, 13.31385302566793651356660744058, 14.74964043303869700742126825607, 15.54018395149442515258646346459, 16.04495064671967522951730817371, 16.81292288933172053374390668519

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