Properties

Degree $4$
Conductor $2160900$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 5-s − 6-s − 8-s + 10-s + 4·13-s − 15-s − 16-s − 6·17-s + 4·19-s + 24-s + 4·26-s + 27-s − 12·29-s − 30-s − 8·31-s − 6·34-s − 2·37-s + 4·38-s − 4·39-s − 40-s − 12·41-s − 8·43-s + 48-s + 6·51-s + 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 0.316·10-s + 1.10·13-s − 0.258·15-s − 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.204·24-s + 0.784·26-s + 0.192·27-s − 2.22·29-s − 0.182·30-s − 1.43·31-s − 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.640·39-s − 0.158·40-s − 1.87·41-s − 1.21·43-s + 0.144·48-s + 0.840·51-s + 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2160900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2160900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.605079590\)
\(L(\frac12)\) \(\approx\) \(1.605079590\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−9.986059657471924385468788702042, −9.250608206973181109736431461020, −8.967446964032693723866299657097, −8.417222032209946627169246397026, −8.383995362944676154961101668235, −7.35221900014169649938140074366, −7.35124769642876327770377469486, −6.59845097648006794754063757717, −6.52865387457126036365243403908, −5.82427301260982121299444137712, −5.58576797438396899624828823568, −5.10093160939929606062172013613, −4.97211040951406850793019324451, −3.94624877839680820158477878890, −3.93115261673725026515899214661, −3.38162092803765581344209629016, −2.75798005298134093626575036806, −1.80167405542405682818644635247, −1.74889819802170966330111700935, −0.45353104890954810353186116842, 0.45353104890954810353186116842, 1.74889819802170966330111700935, 1.80167405542405682818644635247, 2.75798005298134093626575036806, 3.38162092803765581344209629016, 3.93115261673725026515899214661, 3.94624877839680820158477878890, 4.97211040951406850793019324451, 5.10093160939929606062172013613, 5.58576797438396899624828823568, 5.82427301260982121299444137712, 6.52865387457126036365243403908, 6.59845097648006794754063757717, 7.35124769642876327770377469486, 7.35221900014169649938140074366, 8.383995362944676154961101668235, 8.417222032209946627169246397026, 8.967446964032693723866299657097, 9.250608206973181109736431461020, 9.986059657471924385468788702042

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