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Properties

Label 4-117e2-1.1-c1e2-0-0
Degree $4$
Conductor $13689$
Sign $1$
Analytic cond. $0.872822$
Root an. cond. $0.966565$
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  − 2·3-s − 4-s + 9-s + 2·12-s + 5·13-s − 3·16-s + 6·17-s − 25-s + 4·27-s + 15·29-s − 36-s − 10·39-s − 2·43-s + 6·48-s − 13·49-s − 12·51-s − 5·52-s + 6·53-s + 19·61-s + 7·64-s − 6·68-s + 2·75-s − 20·79-s − 11·81-s − 30·87-s + 100-s − 6·101-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 1/3·9-s + 0.577·12-s + 1.38·13-s − 3/4·16-s + 1.45·17-s − 1/5·25-s + 0.769·27-s + 2.78·29-s − 1/6·36-s − 1.60·39-s − 0.304·43-s + 0.866·48-s − 1.85·49-s − 1.68·51-s − 0.693·52-s + 0.824·53-s + 2.43·61-s + 7/8·64-s − 0.727·68-s + 0.230·75-s − 2.25·79-s − 1.22·81-s − 3.21·87-s + 1/10·100-s − 0.597·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13689\)    =    \(3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.872822\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13689,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6751576074\)
\(L(\frac12)\) \(\approx\) \(0.6751576074\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_2$ \( 1 + 2 T + p T^{2} \)
13$C_2$ \( 1 - 5 T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.a_al
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.29.ap_ei
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.a_z
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.41.a_k
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.c_da
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.47.a_ac
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.ag_db
59$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.59.a_abj
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.61.at_hk
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.71.a_n
73$C_2^2$ \( 1 - 47 T^{2} + p^{2} T^{4} \) 2.73.a_abv
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \) 2.83.a_aep
89$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \) 2.89.a_bl
97$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.97.a_ao
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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

−11.35602805723046122206829210450, −10.68389180192618923629940403141, −10.15085227464949750149406598643, −9.830828044174449564877761972139, −8.901118960746985838598890682533, −8.417598580717825745872133683050, −8.033008851926041234750244332490, −6.93008699605088141542062183856, −6.52242754969305111991545394487, −5.89619929848292488548421093648, −5.26921996746328915551854401084, −4.67556083454843451240290976455, −3.85603569322153007789294912365, −2.89344923925415568822832195876, −1.09621016892456542075964254605, 1.09621016892456542075964254605, 2.89344923925415568822832195876, 3.85603569322153007789294912365, 4.67556083454843451240290976455, 5.26921996746328915551854401084, 5.89619929848292488548421093648, 6.52242754969305111991545394487, 6.93008699605088141542062183856, 8.033008851926041234750244332490, 8.417598580717825745872133683050, 8.901118960746985838598890682533, 9.830828044174449564877761972139, 10.15085227464949750149406598643, 10.68389180192618923629940403141, 11.35602805723046122206829210450

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