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Properties

Label 4-1053e2-1.1-c1e2-0-20
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s − 2·7-s + 2·13-s − 2·19-s − 25-s − 4·28-s − 2·31-s + 10·37-s − 5·43-s − 2·49-s + 4·52-s − 17·61-s − 8·64-s + 4·67-s − 2·73-s − 4·76-s − 2·79-s − 4·91-s − 20·97-s − 2·100-s − 8·103-s − 32·109-s + 2·121-s − 4·124-s + 127-s + 131-s + 4·133-s + ⋯
L(s)  = 1  + 4-s − 0.755·7-s + 0.554·13-s − 0.458·19-s − 1/5·25-s − 0.755·28-s − 0.359·31-s + 1.64·37-s − 0.762·43-s − 2/7·49-s + 0.554·52-s − 2.17·61-s − 64-s + 0.488·67-s − 0.234·73-s − 0.458·76-s − 0.225·79-s − 0.419·91-s − 2.03·97-s − 1/5·100-s − 0.788·103-s − 3.06·109-s + 2/11·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.346·133-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: \(1\)
Selberg data: \((4,\ 1108809,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.2.a_ac
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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
17$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.17.a_n
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.c_be
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.29.a_bi
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.c_cc
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.ak_dm
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.41.a_k
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.f_dm
47$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.47.a_af
53$C_2^2$ \( 1 - 83 T^{2} + p^{2} T^{4} \) 2.53.a_adf
59$C_2^2$ \( 1 - 113 T^{2} + p^{2} T^{4} \) 2.59.a_aej
61$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.r_hk
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ae_dy
71$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \) 2.71.a_acz
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.c_co
79$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.c_p
83$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \) 2.83.a_dz
89$C_2^2$ \( 1 - 95 T^{2} + p^{2} T^{4} \) 2.89.a_adr
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−7.84226080002954015968394152703, −7.35578743883212382727349947866, −6.79694043732077416807887103412, −6.65111829928214628671538498508, −6.13105567121200364120291738511, −5.80803249410114836798381157371, −5.31485653202830864615624498231, −4.47013171218302891820738220679, −4.27444291094218043513557477044, −3.50208947171869667367322083913, −3.03951711983640157538543845005, −2.57903880689061774124589329449, −1.91216222893672947830341232036, −1.25229754857589181841889913305, 0, 1.25229754857589181841889913305, 1.91216222893672947830341232036, 2.57903880689061774124589329449, 3.03951711983640157538543845005, 3.50208947171869667367322083913, 4.27444291094218043513557477044, 4.47013171218302891820738220679, 5.31485653202830864615624498231, 5.80803249410114836798381157371, 6.13105567121200364120291738511, 6.65111829928214628671538498508, 6.79694043732077416807887103412, 7.35578743883212382727349947866, 7.84226080002954015968394152703

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