Properties

Label 4-416e2-1.1-c1e2-0-0
Degree $4$
Conductor $173056$
Sign $1$
Analytic cond. $11.0342$
Root an. cond. $1.82257$
Motivic weight $1$
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  − 2·5-s − 2·7-s + 2·9-s − 6·11-s − 6·13-s + 6·19-s + 2·25-s − 12·29-s − 6·31-s + 4·35-s + 6·37-s + 2·41-s + 8·43-s − 4·45-s − 10·47-s + 2·49-s + 12·53-s + 12·55-s − 14·59-s + 28·61-s − 4·63-s + 12·65-s − 10·67-s + 10·71-s + 18·73-s + 12·77-s − 5·81-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.755·7-s + 2/3·9-s − 1.80·11-s − 1.66·13-s + 1.37·19-s + 2/5·25-s − 2.22·29-s − 1.07·31-s + 0.676·35-s + 0.986·37-s + 0.312·41-s + 1.21·43-s − 0.596·45-s − 1.45·47-s + 2/7·49-s + 1.64·53-s + 1.61·55-s − 1.82·59-s + 3.58·61-s − 0.503·63-s + 1.48·65-s − 1.22·67-s + 1.18·71-s + 2.10·73-s + 1.36·77-s − 5/9·81-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(173056\)    =    \(2^{10} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(11.0342\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 173056,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7071213228\)
\(L(\frac12)\) \(\approx\) \(0.7071213228\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
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

−11.43944022927050098461517154926, −10.96173750481516300347425184576, −10.58697352281184241312716992204, −9.905592884860998882103158672412, −9.731584454989350587810646422209, −9.365098295939244004374170985556, −8.758970292404432880813068789228, −7.85093021649198076209079208680, −7.76391438211159487140878230787, −7.22634437943545366626965811223, −7.21131729277581123489759950423, −6.21889772051936640913666958750, −5.58941928845342107136503117268, −5.04094630012188367036566922776, −4.84364698487181914157644063694, −3.67356029911679108549950133231, −3.66367831139067676831793992606, −2.61021113341454942397550605695, −2.18834617534372465151181343882, −0.52370447396557550538899867305, 0.52370447396557550538899867305, 2.18834617534372465151181343882, 2.61021113341454942397550605695, 3.66367831139067676831793992606, 3.67356029911679108549950133231, 4.84364698487181914157644063694, 5.04094630012188367036566922776, 5.58941928845342107136503117268, 6.21889772051936640913666958750, 7.21131729277581123489759950423, 7.22634437943545366626965811223, 7.76391438211159487140878230787, 7.85093021649198076209079208680, 8.758970292404432880813068789228, 9.365098295939244004374170985556, 9.731584454989350587810646422209, 9.905592884860998882103158672412, 10.58697352281184241312716992204, 10.96173750481516300347425184576, 11.43944022927050098461517154926

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