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Properties

Label 4-1053e2-1.1-c1e2-0-3
Degree $4$
Conductor $1108809$
Sign $1$
Analytic cond. $70.6986$
Root an. cond. $2.89969$
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  − 4-s − 2·7-s + 5·13-s − 3·16-s − 2·19-s − 25-s + 2·28-s − 8·31-s − 14·37-s − 2·43-s − 2·49-s − 5·52-s − 8·61-s + 7·64-s + 4·67-s − 11·73-s + 2·76-s + 4·79-s − 10·91-s − 14·97-s + 100-s + 10·103-s + 19·109-s + 6·112-s − 121-s + 8·124-s + 127-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.755·7-s + 1.38·13-s − 3/4·16-s − 0.458·19-s − 1/5·25-s + 0.377·28-s − 1.43·31-s − 2.30·37-s − 0.304·43-s − 2/7·49-s − 0.693·52-s − 1.02·61-s + 7/8·64-s + 0.488·67-s − 1.28·73-s + 0.229·76-s + 0.450·79-s − 1.04·91-s − 1.42·97-s + 1/10·100-s + 0.985·103-s + 1.81·109-s + 0.566·112-s − 0.0909·121-s + 0.718·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1108809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1108809 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1108809\)    =    \(3^{8} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(70.6986\)
Root analytic conductor: \(2.89969\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1108809,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9039221560\)
\(L(\frac12)\) \(\approx\) \(0.9039221560\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.c_g
11$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.11.a_b
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.c_be
23$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.23.a_b
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.29.a_aba
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.i_bq
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.37.o_et
41$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \) 2.41.a_bl
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.c_g
47$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.47.a_ax
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2^2$ \( 1 - 47 T^{2} + p^{2} T^{4} \) 2.59.a_abv
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ae_dy
71$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.71.a_ack
73$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.l_ga
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.ae_ew
83$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.83.a_n
89$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.89.a_b
97$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.97.o_hz
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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

−8.197181330501675665141673318575, −7.60977154216901774065449204587, −7.11027382857471525213903277799, −6.73086013145082045849194340396, −6.36916345589902292982084876513, −5.84621012386870821586842810565, −5.43607919307540313812268854095, −4.95529524168579606439714045444, −4.27975753383973177126436762478, −3.96653175764800044914385568213, −3.31622441231059203090650649418, −3.10762193633946813024728772163, −2.02133099752009122348389326681, −1.63158118972966901111815822115, −0.42685637746686469940578920371, 0.42685637746686469940578920371, 1.63158118972966901111815822115, 2.02133099752009122348389326681, 3.10762193633946813024728772163, 3.31622441231059203090650649418, 3.96653175764800044914385568213, 4.27975753383973177126436762478, 4.95529524168579606439714045444, 5.43607919307540313812268854095, 5.84621012386870821586842810565, 6.36916345589902292982084876513, 6.73086013145082045849194340396, 7.11027382857471525213903277799, 7.60977154216901774065449204587, 8.197181330501675665141673318575

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