Properties

Label 4-33e4-1.1-c0e2-0-0
Degree $4$
Conductor $1185921$
Sign $1$
Analytic cond. $0.295372$
Root an. cond. $0.737212$
Motivic weight $0$
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  + 2·4-s + 3·16-s − 2·25-s + 4·64-s − 4·97-s − 4·100-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 2·4-s + 3·16-s − 2·25-s + 4·64-s − 4·97-s − 4·100-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1185921\)    =    \(3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(0.295372\)
Root analytic conductor: \(0.737212\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1185921,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.655667825\)
\(L(\frac12)\) \(\approx\) \(1.655667825\)
\(L(1)\) 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)$
bad3 \( 1 \)
11 \( 1 \)
good2$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
7$C_2^2$ \( 1 + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_1$ \( ( 1 + T )^{4} \)
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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

−10.21096728128016961756596708127, −10.06721808329148398180358714857, −9.375810229740356501027215040802, −9.355268859818653488796930526917, −8.210923552930852750726430637685, −8.201978185270831678360105863139, −7.86999774936843300799674658882, −7.21026740810000805014704578668, −6.92379717397623357864914329454, −6.66798072466537257219409274280, −5.99374002314788415982377879248, −5.72178608052953435384779413066, −5.44367271914378711259131893485, −4.63826198419024545315704109965, −3.85803134120321147141647257358, −3.68341013037308675448901994764, −2.74495422624022727431092388158, −2.62409618595247610835911859261, −1.80435812220385207565973103636, −1.39050397970367248922835682035, 1.39050397970367248922835682035, 1.80435812220385207565973103636, 2.62409618595247610835911859261, 2.74495422624022727431092388158, 3.68341013037308675448901994764, 3.85803134120321147141647257358, 4.63826198419024545315704109965, 5.44367271914378711259131893485, 5.72178608052953435384779413066, 5.99374002314788415982377879248, 6.66798072466537257219409274280, 6.92379717397623357864914329454, 7.21026740810000805014704578668, 7.86999774936843300799674658882, 8.201978185270831678360105863139, 8.210923552930852750726430637685, 9.355268859818653488796930526917, 9.375810229740356501027215040802, 10.06721808329148398180358714857, 10.21096728128016961756596708127

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