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Properties

Label 4-48e3-1.1-c1e2-0-16
Degree $4$
Conductor $110592$
Sign $-1$
Analytic cond. $7.05144$
Root an. cond. $1.62955$
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  − 3-s + 9-s − 8·19-s − 6·25-s − 27-s − 8·43-s + 10·49-s + 8·57-s − 24·67-s − 12·73-s + 6·75-s + 81-s + 4·97-s − 22·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s − 10·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 8·171-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.83·19-s − 6/5·25-s − 0.192·27-s − 1.21·43-s + 10/7·49-s + 1.05·57-s − 2.93·67-s − 1.40·73-s + 0.692·75-s + 1/9·81-s + 0.406·97-s − 2·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.824·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s − 0.611·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(110592\)    =    \(2^{12} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(7.05144\)
Root analytic conductor: \(1.62955\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 110592,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.31.a_acg
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.a_adm
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−9.135459178577866106156582253082, −8.845217153269251688340339504041, −8.307364586497704611608347360402, −7.70053410584935377052435315584, −7.29985272217308006563169341764, −6.61054148317551050000431564339, −6.22454479196593255565911899862, −5.74113999096305402508732779153, −5.16131025132479647580181400652, −4.33705674891806129410268432907, −4.15818711802762698838760332167, −3.23727185808276312503609132355, −2.34226573768221861491152792937, −1.56691138460626490918528098992, 0, 1.56691138460626490918528098992, 2.34226573768221861491152792937, 3.23727185808276312503609132355, 4.15818711802762698838760332167, 4.33705674891806129410268432907, 5.16131025132479647580181400652, 5.74113999096305402508732779153, 6.22454479196593255565911899862, 6.61054148317551050000431564339, 7.29985272217308006563169341764, 7.70053410584935377052435315584, 8.307364586497704611608347360402, 8.845217153269251688340339504041, 9.135459178577866106156582253082

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