Properties

Label 4-2106e2-1.1-c1e2-0-9
Degree $4$
Conductor $4435236$
Sign $1$
Analytic cond. $282.794$
Root an. cond. $4.10079$
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  + 2-s + 3·5-s + 7-s − 8-s + 3·10-s − 6·11-s − 13-s + 14-s − 16-s − 6·17-s + 4·19-s − 6·22-s + 5·25-s − 26-s − 6·29-s + 4·31-s − 6·34-s + 3·35-s − 14·37-s + 4·38-s − 3·40-s + 43-s − 3·47-s + 7·49-s + 5·50-s − 18·55-s − 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s − 1.80·11-s − 0.277·13-s + 0.267·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s − 1.27·22-s + 25-s − 0.196·26-s − 1.11·29-s + 0.718·31-s − 1.02·34-s + 0.507·35-s − 2.30·37-s + 0.648·38-s − 0.474·40-s + 0.152·43-s − 0.437·47-s + 49-s + 0.707·50-s − 2.42·55-s − 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4435236\)    =    \(2^{2} \cdot 3^{8} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(282.794\)
Root analytic conductor: \(4.10079\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4435236,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.392373550\)
\(L(\frac12)\) \(\approx\) \(2.392373550\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
13$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−9.610904285049721787433692892374, −8.711962847609570933809958708012, −8.649261794348144660180190534618, −8.330576127552605051562552200408, −7.43881914833250116531329591741, −7.34811904040223645978385172471, −7.10655077801468856013375965359, −6.31028082722603212969895373561, −6.05715653733975771558214886617, −5.62669707282006973526322277630, −5.17625283133247283103941897662, −5.14030210678678737482809281514, −4.44445407002124782653118896156, −4.26939362641974464016591380286, −3.24276691934139067756702467121, −3.11577458653625648403268637582, −2.48316605768371302649721145065, −1.97412007666216217210581245952, −1.66240041016730647752147874086, −0.44942685657554804936093753684, 0.44942685657554804936093753684, 1.66240041016730647752147874086, 1.97412007666216217210581245952, 2.48316605768371302649721145065, 3.11577458653625648403268637582, 3.24276691934139067756702467121, 4.26939362641974464016591380286, 4.44445407002124782653118896156, 5.14030210678678737482809281514, 5.17625283133247283103941897662, 5.62669707282006973526322277630, 6.05715653733975771558214886617, 6.31028082722603212969895373561, 7.10655077801468856013375965359, 7.34811904040223645978385172471, 7.43881914833250116531329591741, 8.330576127552605051562552200408, 8.649261794348144660180190534618, 8.711962847609570933809958708012, 9.610904285049721787433692892374

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