Properties

Label 4-684e2-1.1-c2e2-0-3
Degree $4$
Conductor $467856$
Sign $1$
Analytic cond. $347.362$
Root an. cond. $4.31713$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 22·7-s + 45·13-s − 26·19-s + 25·25-s − 83·43-s + 265·49-s − 121·61-s + 231·67-s − 143·73-s + 153·79-s − 990·91-s + 336·97-s + 72·109-s − 242·121-s + 127-s + 131-s + 572·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.18e3·169-s + 173-s − 550·175-s + ⋯
L(s)  = 1  − 3.14·7-s + 3.46·13-s − 1.36·19-s + 25-s − 1.93·43-s + 5.40·49-s − 1.98·61-s + 3.44·67-s − 1.95·73-s + 1.93·79-s − 10.8·91-s + 3.46·97-s + 0.660·109-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 4.30·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 6.98·169-s + 0.00578·173-s − 3.14·175-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(467856\)    =    \(2^{4} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(347.362\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{684} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 467856,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.266019412\)
\(L(\frac12)\) \(\approx\) \(1.266019412\)
\(L(2)\) 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 \( 1 \)
3 \( 1 \)
19$C_2$ \( 1 + 26 T + p^{2} T^{2} \)
good5$C_2^2$ \( 1 - p^{2} T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 + 11 T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 23 T + p^{2} T^{2} )( 1 - 22 T + p^{2} T^{2} ) \)
17$C_2^2$ \( 1 - p^{2} T^{2} + p^{4} T^{4} \)
23$C_2^2$ \( 1 - p^{2} T^{2} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
31$C_2$ \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \)
37$C_2$ \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 47 T + p^{2} T^{2} ) \)
41$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
43$C_2$ \( ( 1 + 22 T + p^{2} T^{2} )( 1 + 61 T + p^{2} T^{2} ) \)
47$C_2^2$ \( 1 - p^{2} T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
59$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
61$C_2$ \( ( 1 + 47 T + p^{2} T^{2} )( 1 + 74 T + p^{2} T^{2} ) \)
67$C_2$ \( ( 1 - 122 T + p^{2} T^{2} )( 1 - 109 T + p^{2} T^{2} ) \)
71$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
73$C_2$ \( ( 1 + 46 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \)
79$C_2$ \( ( 1 - 142 T + p^{2} T^{2} )( 1 - 11 T + p^{2} T^{2} ) \)
83$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
89$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
97$C_2$ \( ( 1 - 169 T + p^{2} T^{2} )( 1 - 167 T + p^{2} T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.43759936367809315199317138670, −10.07134004508359129576049360505, −9.692261486005965048110378838459, −9.094984965812344253487554075684, −8.729617651986676739183317895115, −8.666043772186825028148405398702, −8.023527167147012468724841163045, −7.25627531938221094505396265960, −6.60564286120665850318056002212, −6.40952927209120542109622744322, −6.28018464851400605090039095980, −5.90565837167600327482695798339, −5.12049721063222219240393062631, −4.27690279885097168009028848205, −3.57492036390584089042156084482, −3.53294852899361318296913311927, −3.13300107982041113196853556280, −2.24360501495517763523392124814, −1.22947968115215383432302855546, −0.44371192407274583291046544366, 0.44371192407274583291046544366, 1.22947968115215383432302855546, 2.24360501495517763523392124814, 3.13300107982041113196853556280, 3.53294852899361318296913311927, 3.57492036390584089042156084482, 4.27690279885097168009028848205, 5.12049721063222219240393062631, 5.90565837167600327482695798339, 6.28018464851400605090039095980, 6.40952927209120542109622744322, 6.60564286120665850318056002212, 7.25627531938221094505396265960, 8.023527167147012468724841163045, 8.666043772186825028148405398702, 8.729617651986676739183317895115, 9.094984965812344253487554075684, 9.692261486005965048110378838459, 10.07134004508359129576049360505, 10.43759936367809315199317138670

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