Properties

Label 4-1177344-1.1-c1e2-0-0
Degree $4$
Conductor $1177344$
Sign $1$
Analytic cond. $75.0684$
Root an. cond. $2.94350$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 7-s + 9-s + 2·13-s − 8·19-s + 2·21-s + 6·25-s − 4·27-s + 4·39-s + 6·43-s − 6·49-s − 16·57-s + 63-s + 24·67-s + 3·73-s + 12·75-s − 4·79-s − 11·81-s + 2·91-s + 16·97-s + 20·103-s − 20·109-s + 2·117-s − 14·121-s + 127-s + 12·129-s + 131-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 1.83·19-s + 0.436·21-s + 6/5·25-s − 0.769·27-s + 0.640·39-s + 0.914·43-s − 6/7·49-s − 2.11·57-s + 0.125·63-s + 2.93·67-s + 0.351·73-s + 1.38·75-s − 0.450·79-s − 1.22·81-s + 0.209·91-s + 1.62·97-s + 1.97·103-s − 1.91·109-s + 0.184·117-s − 1.27·121-s + 0.0887·127-s + 1.05·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1177344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1177344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1177344\)    =    \(2^{8} \cdot 3^{2} \cdot 7 \cdot 73\)
Sign: $1$
Analytic conductor: \(75.0684\)
Root analytic conductor: \(2.94350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1177344} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1177344,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.146761764\)
\(L(\frac12)\) \(\approx\) \(3.146761764\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
73$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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

−8.239336093449914678215428399846, −7.77088674661830994616706802953, −7.22073181493104017937857231333, −6.73874007584629692411494077417, −6.39304485958783668461557447120, −5.88928065681144139011770811900, −5.32892045406048220080698236465, −4.79080982183336146712261414882, −4.29402601048030716451673073527, −3.85696710448615271195762637707, −3.33141183645955123887138028271, −2.78794560387640188810932761881, −2.18144682302881447475320892165, −1.78754946425480483953537519008, −0.73688601445894288440962212577, 0.73688601445894288440962212577, 1.78754946425480483953537519008, 2.18144682302881447475320892165, 2.78794560387640188810932761881, 3.33141183645955123887138028271, 3.85696710448615271195762637707, 4.29402601048030716451673073527, 4.79080982183336146712261414882, 5.32892045406048220080698236465, 5.88928065681144139011770811900, 6.39304485958783668461557447120, 6.73874007584629692411494077417, 7.22073181493104017937857231333, 7.77088674661830994616706802953, 8.239336093449914678215428399846

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