Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} $
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 + 5-s − 4·7-s − 8-s + 10-s − 5·11-s − 10·13-s − 4·14-s − 16-s − 4·17-s + 7·19-s − 5·22-s + 23-s − 10·26-s + 2·31-s − 4·34-s − 4·35-s − 37-s + 7·38-s − 40-s − 10·41-s + 24·43-s + 46-s − 11·47-s + 9·49-s − 9·53-s − 5·55-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s − 1.50·11-s − 2.77·13-s − 1.06·14-s − 1/4·16-s − 0.970·17-s + 1.60·19-s − 1.06·22-s + 0.208·23-s − 1.96·26-s + 0.359·31-s − 0.685·34-s − 0.676·35-s − 0.164·37-s + 1.13·38-s − 0.158·40-s − 1.56·41-s + 3.65·43-s + 0.147·46-s − 1.60·47-s + 9/7·49-s − 1.23·53-s − 0.674·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{630} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.919130\)
\(L(\frac12)\)  \(\approx\)  \(0.919130\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 11 T + 74 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.78127876532032434845701648063, −10.18428091831982305473912672176, −9.942536779318481670520373852285, −9.543021933622657216761220812249, −9.348084258811557264347194738507, −8.824237940396154554875506103122, −7.987044870672589076941015267156, −7.53550592955590719540367728156, −7.36647492212686621800083458975, −6.65236343992326758403492498746, −6.40845125194464362529193886438, −5.58756348665567641289927285318, −5.36720858753066298400408584054, −4.76693209639527399363167283380, −4.59177878133704356505154841170, −3.53400530488337618788129251009, −3.08816430600219330754148544304, −2.53113072379842688939792474628, −2.23024409082416265018490320927, −0.43523103010652206430414709348, 0.43523103010652206430414709348, 2.23024409082416265018490320927, 2.53113072379842688939792474628, 3.08816430600219330754148544304, 3.53400530488337618788129251009, 4.59177878133704356505154841170, 4.76693209639527399363167283380, 5.36720858753066298400408584054, 5.58756348665567641289927285318, 6.40845125194464362529193886438, 6.65236343992326758403492498746, 7.36647492212686621800083458975, 7.53550592955590719540367728156, 7.987044870672589076941015267156, 8.824237940396154554875506103122, 9.348084258811557264347194738507, 9.543021933622657216761220812249, 9.942536779318481670520373852285, 10.18428091831982305473912672176, 10.78127876532032434845701648063

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