Properties

Label 4-48e3-1.1-c1e2-0-15
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 − 4·13-s − 8·23-s + 2·25-s − 27-s + 4·37-s + 4·39-s − 8·47-s + 2·49-s − 24·59-s + 4·61-s + 8·69-s − 8·71-s − 12·73-s − 2·75-s + 81-s + 16·83-s − 12·97-s − 8·107-s + 12·109-s − 4·111-s − 4·117-s − 6·121-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.10·13-s − 1.66·23-s + 2/5·25-s − 0.192·27-s + 0.657·37-s + 0.640·39-s − 1.16·47-s + 2/7·49-s − 3.12·59-s + 0.512·61-s + 0.963·69-s − 0.949·71-s − 1.40·73-s − 0.230·75-s + 1/9·81-s + 1.75·83-s − 1.21·97-s − 0.773·107-s + 1.14·109-s − 0.379·111-s − 0.369·117-s − 0.545·121-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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.e_o
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.i_bu
29$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.29.a_ac
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.ae_o
41$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.41.a_aco
43$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.43.a_aba
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.i_dq
53$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.53.a_da
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.67.a_cc
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.i_fm
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.m_eo
79$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.79.a_ade
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.83.aq_ig
89$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.89.a_ade
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.m_gk
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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

−9.432492825322279800274577262691, −8.812968259329658411193749171444, −8.155722127840238427557131262644, −7.72664778986278417190321476459, −7.33266169137427843035147481929, −6.68757513469487254875814581476, −6.13373234655557169625960733669, −5.79891080752653884032363983386, −4.98887377198599915801402269328, −4.62949911584812408799883825932, −4.01664132643974806054010385839, −3.18067323474391559812618011843, −2.41456168053873649562575530957, −1.53461289720815945124585574604, 0, 1.53461289720815945124585574604, 2.41456168053873649562575530957, 3.18067323474391559812618011843, 4.01664132643974806054010385839, 4.62949911584812408799883825932, 4.98887377198599915801402269328, 5.79891080752653884032363983386, 6.13373234655557169625960733669, 6.68757513469487254875814581476, 7.33266169137427843035147481929, 7.72664778986278417190321476459, 8.155722127840238427557131262644, 8.812968259329658411193749171444, 9.432492825322279800274577262691

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