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  − 4-s + 4·5-s − 9-s − 10·11-s + 16-s − 14·19-s − 4·20-s + 11·25-s − 12·31-s + 36-s − 18·41-s + 10·44-s − 4·45-s − 40·55-s − 8·59-s − 4·61-s − 64-s − 4·71-s + 14·76-s + 28·79-s + 4·80-s + 81-s − 20·89-s − 56·95-s + 10·99-s − 11·100-s + 16·101-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s − 1/3·9-s − 3.01·11-s + 1/4·16-s − 3.21·19-s − 0.894·20-s + 11/5·25-s − 2.15·31-s + 1/6·36-s − 2.81·41-s + 1.50·44-s − 0.596·45-s − 5.39·55-s − 1.04·59-s − 0.512·61-s − 1/8·64-s − 0.474·71-s + 1.60·76-s + 3.15·79-s + 0.447·80-s + 1/9·81-s − 2.11·89-s − 5.74·95-s + 1.00·99-s − 1.09·100-s + 1.59·101-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

\(L(1)\) \(\approx\) \(0.3753458665\)
\(L(\frac12)\) \(\approx\) \(0.3753458665\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
7 \( 1 \)
good11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 75 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−9.886854455635435559973941916322, −9.105756100425484653667666449819, −9.027115059790585158195686307713, −8.411092767280935435889115081642, −8.388886862842258079571312699229, −7.70976505900251366710457066509, −7.42151430090843061699936346803, −6.61853468352170945919737778273, −6.48564632037147651937050486620, −5.86435158814388079943570036739, −5.60923192592448998276820419979, −5.02093583675257517395915830129, −5.00908477864696606905056858633, −4.42043626204439815208479484311, −3.59052577848097768589292253766, −3.07332614542698077553300296824, −2.47132488234768424377477649349, −2.00821111096943333565913103016, −1.83193334314084235248106097083, −0.21987276389889351163085446506, 0.21987276389889351163085446506, 1.83193334314084235248106097083, 2.00821111096943333565913103016, 2.47132488234768424377477649349, 3.07332614542698077553300296824, 3.59052577848097768589292253766, 4.42043626204439815208479484311, 5.00908477864696606905056858633, 5.02093583675257517395915830129, 5.60923192592448998276820419979, 5.86435158814388079943570036739, 6.48564632037147651937050486620, 6.61853468352170945919737778273, 7.42151430090843061699936346803, 7.70976505900251366710457066509, 8.388886862842258079571312699229, 8.411092767280935435889115081642, 9.027115059790585158195686307713, 9.105756100425484653667666449819, 9.886854455635435559973941916322

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