Properties

Label 4-4608-1.1-c1e2-0-1
Degree $4$
Conductor $4608$
Sign $1$
Analytic cond. $0.293810$
Root an. cond. $0.736235$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s + 4·17-s − 16·23-s − 6·25-s + 16·31-s − 12·41-s − 14·49-s + 16·71-s + 20·73-s − 16·79-s + 81-s − 12·89-s + 4·97-s + 32·103-s + 36·113-s − 6·121-s + ⋯
L(s)  = 1  + 1/3·9-s + 0.970·17-s − 3.33·23-s − 6/5·25-s + 2.87·31-s − 1.87·41-s − 2·49-s + 1.89·71-s + 2.34·73-s − 1.80·79-s + 1/9·81-s − 1.27·89-s + 0.406·97-s + 3.15·103-s + 3.38·113-s − 0.545·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(0.293810\)
Root analytic conductor: \(0.736235\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4608,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8221105817\)
\(L(\frac12)\) \(\approx\) \(0.8221105817\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.23.q_eg
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
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

−12.31725686074733747032880351584, −11.58676923620628310825913776398, −11.58204746144152243175303945055, −10.20677110069933956172337352464, −9.964671915727173343349228861429, −9.785656483951954927896383611288, −8.456614023795656741730916500766, −8.098990694093691505068092710868, −7.61947548271568794045481490015, −6.42897107072744719896577758454, −6.16826675668887144123811676692, −5.14444399128618341481533912587, −4.25303028692796488061253338187, −3.43299575130251299450107440988, −1.99999348241168201324552330088, 1.99999348241168201324552330088, 3.43299575130251299450107440988, 4.25303028692796488061253338187, 5.14444399128618341481533912587, 6.16826675668887144123811676692, 6.42897107072744719896577758454, 7.61947548271568794045481490015, 8.098990694093691505068092710868, 8.456614023795656741730916500766, 9.785656483951954927896383611288, 9.964671915727173343349228861429, 10.20677110069933956172337352464, 11.58204746144152243175303945055, 11.58676923620628310825913776398, 12.31725686074733747032880351584

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