Properties

Label 4-1012e2-1.1-c1e2-0-0
Degree $4$
Conductor $1024144$
Sign $1$
Analytic cond. $65.3003$
Root an. cond. $2.84268$
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  − 2·2-s + 2·4-s − 4·7-s − 5·9-s + 2·11-s + 8·13-s + 8·14-s − 4·16-s + 10·18-s − 4·22-s − 23-s − 9·25-s − 16·26-s − 8·28-s + 8·32-s − 10·36-s − 16·41-s − 12·43-s + 4·44-s + 2·46-s − 2·49-s + 18·50-s + 16·52-s + 20·63-s − 8·64-s − 14·67-s + 8·73-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.51·7-s − 5/3·9-s + 0.603·11-s + 2.21·13-s + 2.13·14-s − 16-s + 2.35·18-s − 0.852·22-s − 0.208·23-s − 9/5·25-s − 3.13·26-s − 1.51·28-s + 1.41·32-s − 5/3·36-s − 2.49·41-s − 1.82·43-s + 0.603·44-s + 0.294·46-s − 2/7·49-s + 2.54·50-s + 2.21·52-s + 2.51·63-s − 64-s − 1.71·67-s + 0.936·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1024144\)    =    \(2^{4} \cdot 11^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(65.3003\)
Root analytic conductor: \(2.84268\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1024144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1679471276\)
\(L(\frac12)\) \(\approx\) \(0.1679471276\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
23$C_2$ \( 1 + T + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−8.400833678377331338399382179998, −7.904347343163503947239497223655, −7.30539977091509490374991102992, −6.72190583101448840377296528633, −6.36261389471308870138602900888, −6.17641373780313558260987605647, −5.68870592576447690197622018396, −5.14626113993013304092658122814, −4.25646502275072011719856474514, −3.71736014672598711319275125352, −3.24850021595906499346532309133, −2.97090325003680113087997283624, −1.87111490701199696845753046079, −1.45937565887769862597300977659, −0.23963680466745170467196108848, 0.23963680466745170467196108848, 1.45937565887769862597300977659, 1.87111490701199696845753046079, 2.97090325003680113087997283624, 3.24850021595906499346532309133, 3.71736014672598711319275125352, 4.25646502275072011719856474514, 5.14626113993013304092658122814, 5.68870592576447690197622018396, 6.17641373780313558260987605647, 6.36261389471308870138602900888, 6.72190583101448840377296528633, 7.30539977091509490374991102992, 7.904347343163503947239497223655, 8.400833678377331338399382179998

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