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Properties

Label 4-416e2-1.1-c0e2-0-0
Degree $4$
Conductor $173056$
Sign $1$
Analytic cond. $0.0431023$
Root an. cond. $0.455643$
Motivic weight $0$
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  + 2·5-s − 2·9-s + 2·25-s + 2·37-s − 2·41-s − 4·45-s − 4·53-s − 4·61-s + 2·73-s + 3·81-s − 2·89-s + 2·97-s + 2·109-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 2·5-s − 2·9-s + 2·25-s + 2·37-s − 2·41-s − 4·45-s − 4·53-s − 4·61-s + 2·73-s + 3·81-s − 2·89-s + 2·97-s + 2·109-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(173056\)    =    \(2^{10} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.0431023\)
Root analytic conductor: \(0.455643\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 173056,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8342136937\)
\(L(\frac12)\) \(\approx\) \(0.8342136937\)
\(L(1)\) 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 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good3$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$ \( ( 1 + T )^{4} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$ \( ( 1 + T )^{4} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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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

−11.35159825788314744739280946884, −11.31069611276589590934868735387, −10.72184922602421536327586719363, −10.35723704972721346096764210950, −9.566869835608252480345014322619, −9.522346323367501895718308396668, −9.133629002780230953314977305340, −8.511196947928718310135508737938, −8.078498275041504390228503343954, −7.66184863277729996560368845822, −6.71394351084427805407794498246, −6.27893573026064318113826885391, −5.93769959301795202666695387774, −5.72025115057794121051462155357, −4.84197831786097185235208915102, −4.71912387984632983114618621555, −3.16866821116746938307111895430, −3.15044123081260147708106111956, −2.23985557201118057159815943019, −1.63117168589590306032425342219, 1.63117168589590306032425342219, 2.23985557201118057159815943019, 3.15044123081260147708106111956, 3.16866821116746938307111895430, 4.71912387984632983114618621555, 4.84197831786097185235208915102, 5.72025115057794121051462155357, 5.93769959301795202666695387774, 6.27893573026064318113826885391, 6.71394351084427805407794498246, 7.66184863277729996560368845822, 8.078498275041504390228503343954, 8.511196947928718310135508737938, 9.133629002780230953314977305340, 9.522346323367501895718308396668, 9.566869835608252480345014322619, 10.35723704972721346096764210950, 10.72184922602421536327586719363, 11.31069611276589590934868735387, 11.35159825788314744739280946884

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