Properties

Label 4-63e2-1.1-c1e2-0-0
Degree $4$
Conductor $3969$
Sign $1$
Analytic cond. $0.253066$
Root an. cond. $0.709265$
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 − 2·5-s + 3·6-s + 7-s − 5·8-s + 6·9-s + 2·10-s + 10·11-s − 6·12-s + 5·13-s − 14-s + 6·15-s + 5·16-s − 3·17-s − 6·18-s − 19-s − 4·20-s − 3·21-s − 10·22-s + 6·23-s + 15·24-s − 7·25-s − 5·26-s − 9·27-s + 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 4-s − 0.894·5-s + 1.22·6-s + 0.377·7-s − 1.76·8-s + 2·9-s + 0.632·10-s + 3.01·11-s − 1.73·12-s + 1.38·13-s − 0.267·14-s + 1.54·15-s + 5/4·16-s − 0.727·17-s − 1.41·18-s − 0.229·19-s − 0.894·20-s − 0.654·21-s − 2.13·22-s + 1.25·23-s + 3.06·24-s − 7/5·25-s − 0.980·26-s − 1.73·27-s + 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.253066\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{63} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3969,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3866984957\)
\(L(\frac12)\) \(\approx\) \(0.3866984957\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 3 T - 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
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

−15.47674288230236081681088806183, −14.87376368474501683723996129600, −14.51799207339761565762672581044, −13.46385107895923437615042734149, −12.70978823332168147455043032808, −11.86846019163676428235716632122, −11.71708196006409488967224957758, −11.56480729303810202545692114285, −11.02430094162689954314665678942, −10.39171380354547441234432551552, −9.293965850403350358128547671807, −9.058662928250434748634489185049, −8.316946475014102569650881926035, −7.16140752125138540149038860546, −6.71366573090050386876404957274, −6.25357711914196792490783242610, −5.69288996773464311684903806541, −4.24046345121247578862061925991, −3.72236946875674149886478688528, −1.34481939087700784078345092839, 1.34481939087700784078345092839, 3.72236946875674149886478688528, 4.24046345121247578862061925991, 5.69288996773464311684903806541, 6.25357711914196792490783242610, 6.71366573090050386876404957274, 7.16140752125138540149038860546, 8.316946475014102569650881926035, 9.058662928250434748634489185049, 9.293965850403350358128547671807, 10.39171380354547441234432551552, 11.02430094162689954314665678942, 11.56480729303810202545692114285, 11.71708196006409488967224957758, 11.86846019163676428235716632122, 12.70978823332168147455043032808, 13.46385107895923437615042734149, 14.51799207339761565762672581044, 14.87376368474501683723996129600, 15.47674288230236081681088806183

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