Properties

Label 4-645248-1.1-c1e2-0-5
Degree $4$
Conductor $645248$
Sign $-1$
Analytic cond. $41.1415$
Root an. cond. $2.53262$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s + 2·6-s + 8-s − 3·9-s + 2·12-s + 16-s − 3·18-s − 2·19-s + 2·24-s − 10·25-s − 14·27-s + 32-s − 3·36-s − 2·38-s − 2·43-s + 2·48-s − 13·49-s − 10·50-s − 14·54-s − 4·57-s + 12·59-s + 64-s + 16·67-s − 3·72-s − 2·73-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s − 9-s + 0.577·12-s + 1/4·16-s − 0.707·18-s − 0.458·19-s + 0.408·24-s − 2·25-s − 2.69·27-s + 0.176·32-s − 1/2·36-s − 0.324·38-s − 0.304·43-s + 0.288·48-s − 1.85·49-s − 1.41·50-s − 1.90·54-s − 0.529·57-s + 1.56·59-s + 1/8·64-s + 1.95·67-s − 0.353·72-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 645248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 645248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(645248\)    =    \(2^{7} \cdot 71^{2}\)
Sign: $-1$
Analytic conductor: \(41.1415\)
Root analytic conductor: \(2.53262\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 645248,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
71$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
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

−8.155283673687832855649623537912, −7.79510581397524982652968495375, −7.43796356539382262615015727330, −6.52013654011009885219318723813, −6.45900320635304245713021356815, −5.79481497785218672150846122983, −5.30767778357765891574349430596, −5.02377036581249468373225138878, −4.04396073525979748019485118827, −3.77653001984849480071749037204, −3.36885759731685286955180224788, −2.51874373869269853485475383946, −2.41057075071613019915362974603, −1.59657674380372298540849156816, 0, 1.59657674380372298540849156816, 2.41057075071613019915362974603, 2.51874373869269853485475383946, 3.36885759731685286955180224788, 3.77653001984849480071749037204, 4.04396073525979748019485118827, 5.02377036581249468373225138878, 5.30767778357765891574349430596, 5.79481497785218672150846122983, 6.45900320635304245713021356815, 6.52013654011009885219318723813, 7.43796356539382262615015727330, 7.79510581397524982652968495375, 8.155283673687832855649623537912

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