Properties

Degree $4$
Conductor $10497600$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 4·11-s − 6·13-s − 12·17-s − 8·19-s + 2·29-s + 8·31-s − 4·37-s + 6·41-s − 12·43-s − 8·47-s + 7·49-s + 12·53-s − 4·55-s − 12·59-s − 14·61-s + 6·65-s − 4·67-s + 16·71-s − 12·73-s + 8·79-s + 12·83-s + 12·85-s + 20·89-s + 8·95-s − 2·97-s − 6·101-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.20·11-s − 1.66·13-s − 2.91·17-s − 1.83·19-s + 0.371·29-s + 1.43·31-s − 0.657·37-s + 0.937·41-s − 1.82·43-s − 1.16·47-s + 49-s + 1.64·53-s − 0.539·55-s − 1.56·59-s − 1.79·61-s + 0.744·65-s − 0.488·67-s + 1.89·71-s − 1.40·73-s + 0.900·79-s + 1.31·83-s + 1.30·85-s + 2.11·89-s + 0.820·95-s − 0.203·97-s − 0.597·101-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: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3240} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.499466376583209059613860942847, −8.242155141583757194759864805876, −7.62911776757967538617663112329, −7.40600431903669641257903946210, −6.78629005098767373322008202528, −6.50321884540174078969104703187, −6.48506926316086115192751231418, −6.10449497952573634087183106523, −5.14889466791944335456037666305, −4.95848890397936305559578185465, −4.45905685587002289731098115620, −4.31585622469703807855010308937, −3.91663436435070860240344292739, −3.32764672861965829004093677583, −2.58676221891270197606603675293, −2.35199065601340311288318228898, −1.95014402850216305125437303690, −1.19723752386440659352610619767, 0, 0, 1.19723752386440659352610619767, 1.95014402850216305125437303690, 2.35199065601340311288318228898, 2.58676221891270197606603675293, 3.32764672861965829004093677583, 3.91663436435070860240344292739, 4.31585622469703807855010308937, 4.45905685587002289731098115620, 4.95848890397936305559578185465, 5.14889466791944335456037666305, 6.10449497952573634087183106523, 6.48506926316086115192751231418, 6.50321884540174078969104703187, 6.78629005098767373322008202528, 7.40600431903669641257903946210, 7.62911776757967538617663112329, 8.242155141583757194759864805876, 8.499466376583209059613860942847

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