Properties

Degree $4$
Conductor $926100$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s − 2·5-s + 7-s + 9-s + 12-s − 2·15-s + 16-s − 12·17-s − 2·20-s + 21-s + 3·25-s + 27-s + 28-s − 2·35-s + 36-s + 4·37-s + 12·41-s + 16·43-s − 2·45-s − 24·47-s + 48-s + 49-s − 12·51-s − 24·59-s − 2·60-s + 63-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.288·12-s − 0.516·15-s + 1/4·16-s − 2.91·17-s − 0.447·20-s + 0.218·21-s + 3/5·25-s + 0.192·27-s + 0.188·28-s − 0.338·35-s + 1/6·36-s + 0.657·37-s + 1.87·41-s + 2.43·43-s − 0.298·45-s − 3.50·47-s + 0.144·48-s + 1/7·49-s − 1.68·51-s − 3.12·59-s − 0.258·60-s + 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 926100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 926100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(926100\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 7^{3}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{926100} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 926100,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 - T \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( 1 - T \)
good11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−7.991006296285731130234927884691, −7.41880334400769005077113607898, −7.27965583167809929942184453108, −6.66525158720960711652638814741, −6.14620005069006563789988203374, −6.03347639479763526485218429964, −4.88823096770031111535459012199, −4.64849576847169223614699526425, −4.31977202761190735797982138368, −3.71822253537069739639390686069, −3.12590542097473718225449308769, −2.43310152932511024940703619852, −2.16105217917237072156284139202, −1.22061090343879583892735969324, 0, 1.22061090343879583892735969324, 2.16105217917237072156284139202, 2.43310152932511024940703619852, 3.12590542097473718225449308769, 3.71822253537069739639390686069, 4.31977202761190735797982138368, 4.64849576847169223614699526425, 4.88823096770031111535459012199, 6.03347639479763526485218429964, 6.14620005069006563789988203374, 6.66525158720960711652638814741, 7.27965583167809929942184453108, 7.41880334400769005077113607898, 7.991006296285731130234927884691

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