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Properties

Label 4-15984-1.1-c1e2-0-2
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 $1$

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: \(1\)
Selberg data: \((4,\ 15984,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.c_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$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.e_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$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.ag_dq
53$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.53.a_ada
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.q_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$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.o_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$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.m_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

−10.51662787499965966455346695720, −10.19037234413100538344738145697, −9.784237292788443782552004736651, −9.312046710497049700953959541492, −8.662120126420242763972880565007, −7.86766449082484223714067175890, −7.55137359827899381979324127139, −7.19921719019442555575527677414, −6.18524151030053906360695489048, −5.72420700940709745753180925116, −4.73917230711133342706815633638, −4.26275881939721027575112450347, −2.78087112810243158175426733634, −1.84291672591515072963538937221, 0, 1.84291672591515072963538937221, 2.78087112810243158175426733634, 4.26275881939721027575112450347, 4.73917230711133342706815633638, 5.72420700940709745753180925116, 6.18524151030053906360695489048, 7.19921719019442555575527677414, 7.55137359827899381979324127139, 7.86766449082484223714067175890, 8.662120126420242763972880565007, 9.312046710497049700953959541492, 9.784237292788443782552004736651, 10.19037234413100538344738145697, 10.51662787499965966455346695720

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