Properties

Label 4-127008-1.1-c1e2-0-25
Degree $4$
Conductor $127008$
Sign $-1$
Analytic cond. $8.09814$
Root an. cond. $1.68692$
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-s + 4-s + 8-s − 8·13-s + 16-s − 12·17-s − 10·25-s − 8·26-s + 12·29-s + 32-s − 12·34-s + 4·37-s − 12·41-s + 49-s − 10·50-s − 8·52-s − 12·53-s + 12·58-s + 16·61-s + 64-s − 12·68-s + 4·73-s + 4·74-s − 12·82-s + 12·89-s − 20·97-s + 98-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2.21·13-s + 1/4·16-s − 2.91·17-s − 2·25-s − 1.56·26-s + 2.22·29-s + 0.176·32-s − 2.05·34-s + 0.657·37-s − 1.87·41-s + 1/7·49-s − 1.41·50-s − 1.10·52-s − 1.64·53-s + 1.57·58-s + 2.04·61-s + 1/8·64-s − 1.45·68-s + 0.468·73-s + 0.464·74-s − 1.32·82-s + 1.27·89-s − 2.03·97-s + 0.101·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127008 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127008 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(127008\)    =    \(2^{5} \cdot 3^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(8.09814\)
Root analytic conductor: \(1.68692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{127008} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 127008,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
3 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−9.207333736191082013174831123730, −8.686268182491541384374530740513, −7.967527986115067076241988284772, −7.80365099617056766338064266882, −6.84458804347565715458286520604, −6.71310127772875365703449938682, −6.34752255234086455771568318662, −5.31921059379060528260176389250, −5.05202336643031434355439952711, −4.32309265725798888275246837851, −4.19102102405251164216473944262, −3.04185374086143547449059084416, −2.39101862347891857041372548490, −1.98782431119292345428047370492, 0, 1.98782431119292345428047370492, 2.39101862347891857041372548490, 3.04185374086143547449059084416, 4.19102102405251164216473944262, 4.32309265725798888275246837851, 5.05202336643031434355439952711, 5.31921059379060528260176389250, 6.34752255234086455771568318662, 6.71310127772875365703449938682, 6.84458804347565715458286520604, 7.80365099617056766338064266882, 7.967527986115067076241988284772, 8.686268182491541384374530740513, 9.207333736191082013174831123730

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