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Properties

Label 4-15984-1.1-c1e2-0-5
Degree $4$
Conductor $15984$
Sign $1$
Analytic cond. $1.01915$
Root an. cond. $1.00475$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s + 2·6-s + 9-s + 2·11-s + 2·12-s − 6·13-s − 4·16-s + 2·18-s + 4·22-s + 4·23-s − 8·25-s − 12·26-s + 27-s − 8·32-s + 2·33-s + 2·36-s + 7·37-s − 6·39-s + 4·44-s + 8·46-s − 6·47-s − 4·48-s − 2·49-s − 16·50-s − 12·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1/3·9-s + 0.603·11-s + 0.577·12-s − 1.66·13-s − 16-s + 0.471·18-s + 0.852·22-s + 0.834·23-s − 8/5·25-s − 2.35·26-s + 0.192·27-s − 1.41·32-s + 0.348·33-s + 1/3·36-s + 1.15·37-s − 0.960·39-s + 0.603·44-s + 1.17·46-s − 0.875·47-s − 0.577·48-s − 2/7·49-s − 2.26·50-s − 1.66·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15984\)    =    \(2^{4} \cdot 3^{3} \cdot 37\)
Sign: $1$
Analytic conductor: \(1.01915\)
Root analytic conductor: \(1.00475\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15984,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.415312400\)
\(L(\frac12)\) \(\approx\) \(2.415312400\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 - p T + p T^{2} \)
3$C_1$ \( 1 - T \)
37$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 6 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ac_w
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.g_ba
17$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.17.a_aq
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.ae_o
29$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \) 2.29.a_bg
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.41.a_ac
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.g_dq
53$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.53.a_ada
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.59.aq_gk
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.67.a_ac
71$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.71.ao_ha
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.79.a_acc
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.83.am_gk
89$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.89.a_cy
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.ag_gw
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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

−11.51390034290469963226382068292, −10.60083521200708121538115458563, −9.864813968111050882216270619021, −9.509977471995273355288168996349, −9.008581739336297841102438282569, −8.165032118802819869360562449283, −7.60022683853724191060494204350, −6.95055486066279444921941691112, −6.45091122767320739837289309697, −5.62165678180636810146710015853, −5.03625298630083681112372362801, −4.36261769625640373012786521546, −3.76286605801315940252961939479, −2.89020270814833009968757885068, −2.13582771594787895067922684351, 2.13582771594787895067922684351, 2.89020270814833009968757885068, 3.76286605801315940252961939479, 4.36261769625640373012786521546, 5.03625298630083681112372362801, 5.62165678180636810146710015853, 6.45091122767320739837289309697, 6.95055486066279444921941691112, 7.60022683853724191060494204350, 8.165032118802819869360562449283, 9.008581739336297841102438282569, 9.509977471995273355288168996349, 9.864813968111050882216270619021, 10.60083521200708121538115458563, 11.51390034290469963226382068292

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