Properties

Label 4-3e2-1.1-c8e2-0-0
Degree $4$
Conductor $9$
Sign $1$
Analytic cond. $1.49361$
Root an. cond. $1.10550$
Motivic weight $8$
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  + 90·3-s + 8·4-s − 3.50e3·7-s + 1.53e3·9-s + 720·12-s + 5.14e4·13-s − 6.54e4·16-s + 3.78e4·19-s − 3.15e5·21-s + 7.30e5·25-s − 4.51e5·27-s − 2.80e4·28-s − 7.02e5·31-s + 1.23e4·36-s + 2.67e6·37-s + 4.63e6·39-s − 7.05e6·43-s − 5.89e6·48-s − 2.34e6·49-s + 4.11e5·52-s + 3.40e6·57-s + 1.50e6·61-s − 5.38e6·63-s − 1.04e6·64-s + 4.53e6·67-s + 5.53e7·73-s + 6.57e7·75-s + ⋯
L(s)  = 1  + 10/9·3-s + 1/32·4-s − 1.45·7-s + 0.234·9-s + 0.0347·12-s + 1.80·13-s − 0.999·16-s + 0.290·19-s − 1.61·21-s + 1.87·25-s − 0.850·27-s − 0.0455·28-s − 0.761·31-s + 0.00733·36-s + 1.42·37-s + 2.00·39-s − 2.06·43-s − 1.11·48-s − 0.406·49-s + 0.0563·52-s + 0.322·57-s + 0.108·61-s − 0.341·63-s − 0.0624·64-s + 0.225·67-s + 1.94·73-s + 2.07·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(1.49361\)
Root analytic conductor: \(1.10550\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9,\ (\ :4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.382913471\)
\(L(\frac12)\) \(\approx\) \(1.382913471\)
\(L(5)\) 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$C_2$ \( 1 - 10 p^{2} T + p^{8} T^{2} \)
good2$C_2^2$ \( 1 - p^{3} T^{2} + p^{16} T^{4} \)
5$C_2^2$ \( 1 - 29234 p^{2} T^{2} + p^{16} T^{4} \)
7$C_2$ \( ( 1 + 250 p T + p^{8} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 380283362 T^{2} + p^{16} T^{4} \)
13$C_2$ \( ( 1 - 25730 T + p^{8} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 8342551298 T^{2} + p^{16} T^{4} \)
19$C_2$ \( ( 1 - 18938 T + p^{8} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 64711613182 T^{2} + p^{16} T^{4} \)
29$C_2^2$ \( 1 - 788066452322 T^{2} + p^{16} T^{4} \)
31$C_2$ \( ( 1 + 11338 p T + p^{8} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 1335170 T + p^{8} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 12452468931842 T^{2} + p^{16} T^{4} \)
43$C_2$ \( ( 1 + 3526150 T + p^{8} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30967680304898 T^{2} + p^{16} T^{4} \)
53$C_2^2$ \( 1 - 80936075395298 T^{2} + p^{16} T^{4} \)
59$C_2^2$ \( 1 - 105562517046242 T^{2} + p^{16} T^{4} \)
61$C_2$ \( ( 1 - 753602 T + p^{8} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2268890 T + p^{8} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 1001758688017922 T^{2} + p^{16} T^{4} \)
73$C_2$ \( ( 1 - 27672770 T + p^{8} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 22980982 T + p^{8} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 2352070843223138 T^{2} + p^{16} T^{4} \)
89$C_2^2$ \( 1 - 2600204109557762 T^{2} + p^{16} T^{4} \)
97$C_2$ \( ( 1 - 147271010 T + p^{8} T^{2} )^{2} \)
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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

−25.48457455056030862586971118301, −25.43404712791207914444593543883, −24.24988930999681135601755894035, −23.03125818036933381721165597225, −22.61633771030336282354266566175, −21.40363457458476111200117821823, −20.23187960048157825678945100376, −20.12216169702305363155548438682, −18.82264965358014825023913936402, −18.29684445410031745836940228720, −16.52349202435110678096070736482, −15.90659526368572752200092385994, −14.76513444245845090232475090568, −13.54027667979888221166920257283, −12.97742864252240100208929908279, −11.13538667100853805576196445480, −9.509896719961398489491287140507, −8.537722471016272724437647432597, −6.56108305736119635078666434601, −3.29129434150346804494000997655, 3.29129434150346804494000997655, 6.56108305736119635078666434601, 8.537722471016272724437647432597, 9.509896719961398489491287140507, 11.13538667100853805576196445480, 12.97742864252240100208929908279, 13.54027667979888221166920257283, 14.76513444245845090232475090568, 15.90659526368572752200092385994, 16.52349202435110678096070736482, 18.29684445410031745836940228720, 18.82264965358014825023913936402, 20.12216169702305363155548438682, 20.23187960048157825678945100376, 21.40363457458476111200117821823, 22.61633771030336282354266566175, 23.03125818036933381721165597225, 24.24988930999681135601755894035, 25.43404712791207914444593543883, 25.48457455056030862586971118301

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