Properties

Label 4-406e2-1.1-c1e2-0-12
Degree $4$
Conductor $164836$
Sign $-1$
Analytic cond. $10.5100$
Root an. cond. $1.80053$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 2·7-s + 4·8-s − 5·9-s − 6·11-s − 4·14-s + 5·16-s − 10·18-s − 12·22-s + 8·23-s − 9·25-s − 6·28-s − 2·29-s + 6·32-s − 15·36-s + 16·37-s − 22·43-s − 18·44-s + 16·46-s − 3·49-s − 18·50-s − 22·53-s − 8·56-s − 4·58-s + 10·63-s + 7·64-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.755·7-s + 1.41·8-s − 5/3·9-s − 1.80·11-s − 1.06·14-s + 5/4·16-s − 2.35·18-s − 2.55·22-s + 1.66·23-s − 9/5·25-s − 1.13·28-s − 0.371·29-s + 1.06·32-s − 5/2·36-s + 2.63·37-s − 3.35·43-s − 2.71·44-s + 2.35·46-s − 3/7·49-s − 2.54·50-s − 3.02·53-s − 1.06·56-s − 0.525·58-s + 1.25·63-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164836 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164836 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(164836\)    =    \(2^{2} \cdot 7^{2} \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(10.5100\)
Root analytic conductor: \(1.80053\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 164836,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
7$C_2$ \( 1 + 2 T + p T^{2} \)
29$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−9.021569234772237196506092397028, −8.237914033781111929312629804446, −7.81821112633712404756766747748, −7.71436185563727090632996745083, −6.76381712126143531253349126939, −6.16227613649540161988088922715, −6.11658999858863662469236119804, −5.25136790623537531107690201816, −5.11050011527156459020994869479, −4.49852693974322508959372476460, −3.49161543470772390818361140602, −3.07071881135695746551942332895, −2.79184388792227261783186696974, −1.93407659453531378223231825006, 0, 1.93407659453531378223231825006, 2.79184388792227261783186696974, 3.07071881135695746551942332895, 3.49161543470772390818361140602, 4.49852693974322508959372476460, 5.11050011527156459020994869479, 5.25136790623537531107690201816, 6.11658999858863662469236119804, 6.16227613649540161988088922715, 6.76381712126143531253349126939, 7.71436185563727090632996745083, 7.81821112633712404756766747748, 8.237914033781111929312629804446, 9.021569234772237196506092397028

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