Properties

Label 4-2850e2-1.1-c1e2-0-25
Degree $4$
Conductor $8122500$
Sign $1$
Analytic cond. $517.897$
Root an. cond. $4.77046$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s + 2·11-s + 6·12-s − 4·13-s + 5·16-s − 8·17-s − 6·18-s − 2·19-s − 4·22-s − 2·23-s − 8·24-s + 8·26-s + 4·27-s − 14·29-s − 2·31-s − 6·32-s + 4·33-s + 16·34-s + 9·36-s − 20·37-s + 4·38-s − 8·39-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 1.41·8-s + 9-s + 0.603·11-s + 1.73·12-s − 1.10·13-s + 5/4·16-s − 1.94·17-s − 1.41·18-s − 0.458·19-s − 0.852·22-s − 0.417·23-s − 1.63·24-s + 1.56·26-s + 0.769·27-s − 2.59·29-s − 0.359·31-s − 1.06·32-s + 0.696·33-s + 2.74·34-s + 3/2·36-s − 3.28·37-s + 0.648·38-s − 1.28·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8122500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(517.897\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8122500,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 14 T + 97 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 23 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 112 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 10 T + 121 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 2 T + 113 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 136 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 2 T + 119 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T + 139 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 20 T + 204 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
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

−8.601066503512158468956862718030, −8.487150237271871932613429286926, −7.74741177340978697059668090317, −7.66345948291603500129618933337, −7.26430638784424561037313504039, −6.77958575296109261392586491330, −6.65591232676171535683224167358, −6.22547768805342446486454866887, −5.38391687667509678727337298964, −5.33008398925280759863601443405, −4.38928191573525178942260395744, −4.29604835594538662977575648789, −3.53545487256366234183197361891, −3.33188453095709915792468874551, −2.63063946994504521055797895436, −2.09778565179684852817163246333, −1.86256759899833465412097939838, −1.48473002893273732988420711179, 0, 0, 1.48473002893273732988420711179, 1.86256759899833465412097939838, 2.09778565179684852817163246333, 2.63063946994504521055797895436, 3.33188453095709915792468874551, 3.53545487256366234183197361891, 4.29604835594538662977575648789, 4.38928191573525178942260395744, 5.33008398925280759863601443405, 5.38391687667509678727337298964, 6.22547768805342446486454866887, 6.65591232676171535683224167358, 6.77958575296109261392586491330, 7.26430638784424561037313504039, 7.66345948291603500129618933337, 7.74741177340978697059668090317, 8.487150237271871932613429286926, 8.601066503512158468956862718030

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