Properties

Label 4-2624e2-1.1-c0e2-0-0
Degree $4$
Conductor $6885376$
Sign $1$
Analytic cond. $1.71491$
Root an. cond. $1.14435$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·25-s + 2·41-s + 4·61-s − 81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 2·25-s + 2·41-s + 4·61-s − 81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6885376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6885376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6885376\)    =    \(2^{12} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(1.71491\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2624} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6885376,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.207253334\)
\(L(\frac12)\) \(\approx\) \(1.207253334\)
\(L(1)\) 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)$
bad2 \( 1 \)
41$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$ \( ( 1 - T )^{4} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−9.280907958022290083985316068125, −8.861233714949802991324685644457, −8.375614469347444373140465859671, −8.179026272766001121000740546663, −7.74554474997688457206723205875, −7.27330438889949300786059855071, −7.13526760690638206462475322907, −6.48914747921063413796135061090, −6.17861051868916698730733355412, −5.75697920079382410204325177622, −5.38050581067249907645391772889, −5.08480677569838671492988258508, −4.21953390371656761064663233021, −4.21247143803608509772161973167, −3.69504420127389968519596264318, −3.18749595835430474944273979542, −2.43882190169921353977936065846, −2.26655906144415210864048456258, −1.54370962335477847069710564947, −0.73713795966271918933991058572, 0.73713795966271918933991058572, 1.54370962335477847069710564947, 2.26655906144415210864048456258, 2.43882190169921353977936065846, 3.18749595835430474944273979542, 3.69504420127389968519596264318, 4.21247143803608509772161973167, 4.21953390371656761064663233021, 5.08480677569838671492988258508, 5.38050581067249907645391772889, 5.75697920079382410204325177622, 6.17861051868916698730733355412, 6.48914747921063413796135061090, 7.13526760690638206462475322907, 7.27330438889949300786059855071, 7.74554474997688457206723205875, 8.179026272766001121000740546663, 8.375614469347444373140465859671, 8.861233714949802991324685644457, 9.280907958022290083985316068125

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