Properties

Label 4-21e2-1.1-c14e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $681.684$
Root an. cond. $5.10970$
Motivic weight $14$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.18e3·3-s − 1.63e4·4-s + 1.46e6·7-s − 3.58e7·12-s − 2.42e8·13-s + 1.58e9·19-s + 3.19e9·21-s − 6.10e9·25-s − 1.04e10·27-s − 2.39e10·28-s − 5.47e10·31-s + 7.71e10·37-s − 5.31e11·39-s − 7.51e11·43-s + 1.45e12·49-s + 3.98e12·52-s + 3.45e12·57-s − 6.21e12·61-s + 4.39e12·64-s − 7.62e12·67-s + 2.05e13·73-s − 1.33e13·75-s − 2.59e13·76-s + 1.79e13·79-s − 2.28e13·81-s − 5.23e13·84-s − 3.54e14·91-s + ⋯
L(s)  = 1  + 3-s − 4-s + 1.77·7-s − 12-s − 3.87·13-s + 1.76·19-s + 1.77·21-s − 25-s − 27-s − 1.77·28-s − 1.99·31-s + 0.813·37-s − 3.87·39-s − 2.76·43-s + 2.14·49-s + 3.87·52-s + 1.76·57-s − 1.97·61-s + 64-s − 1.25·67-s + 1.86·73-s − 75-s − 1.76·76-s + 0.937·79-s − 81-s − 1.77·84-s − 6.86·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+7)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(681.684\)
Root analytic conductor: \(5.10970\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :7, 7),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.9324544142\)
\(L(\frac12)\) \(\approx\) \(0.9324544142\)
\(L(8)\) 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)$
bad3$C_2$ \( 1 - p^{7} T + p^{14} T^{2} \)
7$C_2$ \( 1 - 1461083 T + p^{14} T^{2} \)
good2$C_2$ \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \)
5$C_2$ \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \)
11$C_2$ \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \)
13$C_2$ \( ( 1 + 121460953 T + p^{14} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \)
19$C_2$ \( ( 1 - 1512529226 T + p^{14} T^{2} )( 1 - 69090803 T + p^{14} T^{2} ) \)
23$C_2$ \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
31$C_2$ \( ( 1 + 23014293166 T + p^{14} T^{2} )( 1 + 31777822381 T + p^{14} T^{2} ) \)
37$C_2$ \( ( 1 - 188820174863 T + p^{14} T^{2} )( 1 + 111626070166 T + p^{14} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
43$C_2$ \( ( 1 + 375951067189 T + p^{14} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \)
53$C_2$ \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \)
59$C_2$ \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \)
61$C_2$ \( ( 1 + 2292094939633 T + p^{14} T^{2} )( 1 + 3922504906441 T + p^{14} T^{2} ) \)
67$C_2$ \( ( 1 - 4349633776379 T + p^{14} T^{2} )( 1 + 11973142765462 T + p^{14} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
73$C_2$ \( ( 1 - 17278443497906 T + p^{14} T^{2} )( 1 - 3286657037807 T + p^{14} T^{2} ) \)
79$C_2$ \( ( 1 - 38383122173618 T + p^{14} T^{2} )( 1 + 20384167559989 T + p^{14} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \)
89$C_2$ \( ( 1 - p^{7} T + p^{14} T^{2} )( 1 + p^{7} T + p^{14} T^{2} ) \)
97$C_2$ \( ( 1 + 11651694897022 T + p^{14} T^{2} )^{2} \)
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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

−14.83215135356337646863795034251, −14.55052987733822652218632969001, −13.96000829728448000187573160481, −13.55834315216361054143420088447, −12.49201118215143776327394975258, −11.86848389838022376642984449236, −11.39827862870058565008865776916, −10.12740420026549045311705981592, −9.486815041275943345771137771842, −9.216307836780991248937985454407, −8.145085907925370164012776881926, −7.58747180509224512014421586391, −7.32786783215246874125049580779, −5.22904328605593808585319092764, −5.12167623335907462726605213493, −4.38740147548638926545377930031, −3.22678512779489445760984960249, −2.27371993257418845985672106253, −1.76092814123771531956549913328, −0.28670503647325085913097830634, 0.28670503647325085913097830634, 1.76092814123771531956549913328, 2.27371993257418845985672106253, 3.22678512779489445760984960249, 4.38740147548638926545377930031, 5.12167623335907462726605213493, 5.22904328605593808585319092764, 7.32786783215246874125049580779, 7.58747180509224512014421586391, 8.145085907925370164012776881926, 9.216307836780991248937985454407, 9.486815041275943345771137771842, 10.12740420026549045311705981592, 11.39827862870058565008865776916, 11.86848389838022376642984449236, 12.49201118215143776327394975258, 13.55834315216361054143420088447, 13.96000829728448000187573160481, 14.55052987733822652218632969001, 14.83215135356337646863795034251

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