Properties

Label 4-371e2-1.1-c1e2-0-4
Degree $4$
Conductor $137641$
Sign $-1$
Analytic cond. $8.77610$
Root an. cond. $1.72117$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 7-s + 4·9-s + 4·11-s − 7·13-s − 7·17-s + 2·25-s − 2·28-s − 3·29-s − 8·36-s − 6·37-s − 4·43-s − 8·44-s − 7·47-s − 6·49-s + 14·52-s − 9·53-s − 7·59-s + 4·63-s + 8·64-s + 14·68-s + 4·77-s + 7·81-s + 7·89-s − 7·91-s + 7·97-s + 16·99-s + ⋯
L(s)  = 1  − 4-s + 0.377·7-s + 4/3·9-s + 1.20·11-s − 1.94·13-s − 1.69·17-s + 2/5·25-s − 0.377·28-s − 0.557·29-s − 4/3·36-s − 0.986·37-s − 0.609·43-s − 1.20·44-s − 1.02·47-s − 6/7·49-s + 1.94·52-s − 1.23·53-s − 0.911·59-s + 0.503·63-s + 64-s + 1.69·68-s + 0.455·77-s + 7/9·81-s + 0.741·89-s − 0.733·91-s + 0.710·97-s + 1.60·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(137641\)    =    \(7^{2} \cdot 53^{2}\)
Sign: $-1$
Analytic conductor: \(8.77610\)
Root analytic conductor: \(1.72117\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 137641,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad7$C_2$ \( 1 - T + p T^{2} \)
53$C_2$ \( 1 + 9 T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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

−9.123464599039727574412254112667, −8.878558082970661771699775238836, −8.136208381278717967532960881738, −7.62275587263343131877880006627, −7.12574955735069154540285203212, −6.61721018235880788371758471715, −6.38602252386875560510294136406, −5.14349971255565349607745065325, −4.92127655284803363739289981564, −4.45170359681593866525659157731, −4.07290051039862053056482493100, −3.26489009723204576040084360655, −2.17961200216846533417827059081, −1.56983437907093150983269767856, 0, 1.56983437907093150983269767856, 2.17961200216846533417827059081, 3.26489009723204576040084360655, 4.07290051039862053056482493100, 4.45170359681593866525659157731, 4.92127655284803363739289981564, 5.14349971255565349607745065325, 6.38602252386875560510294136406, 6.61721018235880788371758471715, 7.12574955735069154540285203212, 7.62275587263343131877880006627, 8.136208381278717967532960881738, 8.878558082970661771699775238836, 9.123464599039727574412254112667

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