Properties

Label 4-21e4-1.1-c1e2-0-14
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $12.4002$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s + 2·4-s + 5-s − 3·6-s − 5·8-s + 6·9-s − 10-s − 5·11-s + 6·12-s + 5·13-s + 3·15-s + 5·16-s + 6·17-s − 6·18-s + 2·19-s + 2·20-s + 5·22-s − 3·23-s − 15·24-s + 5·25-s − 5·26-s + 9·27-s + 29-s − 3·30-s − 10·32-s − 15·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 4-s + 0.447·5-s − 1.22·6-s − 1.76·8-s + 2·9-s − 0.316·10-s − 1.50·11-s + 1.73·12-s + 1.38·13-s + 0.774·15-s + 5/4·16-s + 1.45·17-s − 1.41·18-s + 0.458·19-s + 0.447·20-s + 1.06·22-s − 0.625·23-s − 3.06·24-s + 25-s − 0.980·26-s + 1.73·27-s + 0.185·29-s − 0.547·30-s − 1.76·32-s − 2.61·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(12.4002\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.706889470\)
\(L(\frac12)\) \(\approx\) \(2.706889470\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 - p T + p T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
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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

−11.44199195547862612491244305317, −10.68090522209704212163954401593, −10.13952899283317709229470755890, −10.09161808727024984502414164427, −9.343613829796934184959379273249, −9.186340555114057769775308177608, −8.466266875980958445262219159649, −8.373622721455602879529779122248, −7.65432385874067324039912309646, −7.61032563798035430395123958033, −6.88490580498775122344322666374, −6.16945025978157977694502087977, −5.88620763651692772313110837188, −5.28579541960994896071834260574, −4.38542355179131298207500646316, −3.49976096408585882725966553037, −3.06479002676160563166826580188, −2.76840415532828281609946102759, −2.01790317709200663019573455969, −1.16022572186823783108352389968, 1.16022572186823783108352389968, 2.01790317709200663019573455969, 2.76840415532828281609946102759, 3.06479002676160563166826580188, 3.49976096408585882725966553037, 4.38542355179131298207500646316, 5.28579541960994896071834260574, 5.88620763651692772313110837188, 6.16945025978157977694502087977, 6.88490580498775122344322666374, 7.61032563798035430395123958033, 7.65432385874067324039912309646, 8.373622721455602879529779122248, 8.466266875980958445262219159649, 9.186340555114057769775308177608, 9.343613829796934184959379273249, 10.09161808727024984502414164427, 10.13952899283317709229470755890, 10.68090522209704212163954401593, 11.44199195547862612491244305317

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