Properties

Label 4-63e2-1.1-c1e2-0-2
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  + 4-s − 3·5-s + 4·7-s − 3·9-s + 3·11-s + 3·13-s − 3·16-s − 3·17-s − 9·19-s − 3·20-s + 9·23-s + 5·25-s + 4·28-s − 9·29-s − 12·35-s − 3·36-s − 7·37-s − 3·41-s − 43-s + 3·44-s + 9·45-s + 9·49-s + 3·52-s + 15·53-s − 9·55-s − 12·63-s − 7·64-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.34·5-s + 1.51·7-s − 9-s + 0.904·11-s + 0.832·13-s − 3/4·16-s − 0.727·17-s − 2.06·19-s − 0.670·20-s + 1.87·23-s + 25-s + 0.755·28-s − 1.67·29-s − 2.02·35-s − 1/2·36-s − 1.15·37-s − 0.468·41-s − 0.152·43-s + 0.452·44-s + 1.34·45-s + 9/7·49-s + 0.416·52-s + 2.06·53-s − 1.21·55-s − 1.51·63-s − 7/8·64-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

\(L(1)\) \(\approx\) \(0.7803163189\)
\(L(\frac12)\) \(\approx\) \(0.7803163189\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 15 T + 128 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
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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

−15.35751208257009218854921448646, −14.70834666301896751611776578217, −14.56036387730836949818660069239, −13.49058812877495325330719049175, −13.21466091278928643433000240031, −12.16359412649119906651873937512, −11.77754183452763774526465769412, −11.28993963185115458104612883538, −10.83313545070113208300751799377, −10.77707182011915899439555582822, −8.959772757758035218124934610671, −8.824342011500182348229732239823, −8.380777143027558262929981240096, −7.49096196586189799249555304195, −6.91967742670604685606408541649, −6.20075781517012772590908988763, −5.09613453097170482446945321184, −4.34580941512952108439354401796, −3.57904951207528452812854647994, −2.07546201997506410131373966669, 2.07546201997506410131373966669, 3.57904951207528452812854647994, 4.34580941512952108439354401796, 5.09613453097170482446945321184, 6.20075781517012772590908988763, 6.91967742670604685606408541649, 7.49096196586189799249555304195, 8.380777143027558262929981240096, 8.824342011500182348229732239823, 8.959772757758035218124934610671, 10.77707182011915899439555582822, 10.83313545070113208300751799377, 11.28993963185115458104612883538, 11.77754183452763774526465769412, 12.16359412649119906651873937512, 13.21466091278928643433000240031, 13.49058812877495325330719049175, 14.56036387730836949818660069239, 14.70834666301896751611776578217, 15.35751208257009218854921448646

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