Properties

Label 4-3240e2-1.1-c1e2-0-14
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $669.336$
Root an. cond. $5.08640$
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  + 5-s − 2·7-s + 11-s − 13-s + 2·17-s + 8·19-s + 23-s − 5·29-s − 31-s − 2·35-s + 12·37-s − 7·43-s + 7·47-s + 7·49-s + 24·53-s + 55-s − 4·59-s − 10·61-s − 65-s + 4·67-s − 24·71-s + 12·73-s − 2·77-s − 15·79-s + 2·83-s + 2·85-s + 24·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 0.301·11-s − 0.277·13-s + 0.485·17-s + 1.83·19-s + 0.208·23-s − 0.928·29-s − 0.179·31-s − 0.338·35-s + 1.97·37-s − 1.06·43-s + 1.02·47-s + 49-s + 3.29·53-s + 0.134·55-s − 0.520·59-s − 1.28·61-s − 0.124·65-s + 0.488·67-s − 2.84·71-s + 1.40·73-s − 0.227·77-s − 1.68·79-s + 0.219·83-s + 0.216·85-s + 2.54·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(669.336\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3240} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.741263987\)
\(L(\frac12)\) \(\approx\) \(2.741263987\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
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

−8.978996037818273424428764250109, −8.540223972768996793226948795706, −8.030931578743356619696315012254, −7.59762911680978690345969509947, −7.31804891056379523800901404089, −7.13283953037680506822708484828, −6.55536425793309974024182979886, −6.19879515558044651203530969823, −5.71957074898778087340546010500, −5.47748566771949301448685266056, −5.25117483585517765213115052153, −4.55833196703981417968026599152, −4.03304808304314484724377918909, −3.85022534545496910170789524940, −3.13818904541381699451621627970, −2.86739079588172408049261016459, −2.44945930128364634824539687876, −1.71498687479888353872801320952, −1.14665309631874332067934706220, −0.56611576397386358411542041566, 0.56611576397386358411542041566, 1.14665309631874332067934706220, 1.71498687479888353872801320952, 2.44945930128364634824539687876, 2.86739079588172408049261016459, 3.13818904541381699451621627970, 3.85022534545496910170789524940, 4.03304808304314484724377918909, 4.55833196703981417968026599152, 5.25117483585517765213115052153, 5.47748566771949301448685266056, 5.71957074898778087340546010500, 6.19879515558044651203530969823, 6.55536425793309974024182979886, 7.13283953037680506822708484828, 7.31804891056379523800901404089, 7.59762911680978690345969509947, 8.030931578743356619696315012254, 8.540223972768996793226948795706, 8.978996037818273424428764250109

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