Properties

Label 4-12e4-1.1-c1e2-0-10
Degree $4$
Conductor $20736$
Sign $1$
Analytic cond. $1.32214$
Root an. cond. $1.07230$
Motivic weight $1$
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  + 3·3-s − 6·5-s + 6·7-s + 6·9-s + 3·11-s − 4·13-s − 18·15-s + 18·21-s + 19·25-s + 9·27-s − 6·29-s + 9·33-s − 36·35-s + 4·37-s − 12·39-s − 9·41-s − 9·43-s − 36·45-s + 12·47-s + 17·49-s − 18·55-s + 15·59-s − 8·61-s + 36·63-s + 24·65-s + 15·67-s + 12·71-s + ⋯
L(s)  = 1  + 1.73·3-s − 2.68·5-s + 2.26·7-s + 2·9-s + 0.904·11-s − 1.10·13-s − 4.64·15-s + 3.92·21-s + 19/5·25-s + 1.73·27-s − 1.11·29-s + 1.56·33-s − 6.08·35-s + 0.657·37-s − 1.92·39-s − 1.40·41-s − 1.37·43-s − 5.36·45-s + 1.75·47-s + 17/7·49-s − 2.42·55-s + 1.95·59-s − 1.02·61-s + 4.53·63-s + 2.97·65-s + 1.83·67-s + 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(20736\)    =    \(2^{8} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.32214\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{144} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 20736,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.649587999\)
\(L(\frac12)\) \(\approx\) \(1.649587999\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - p T + p T^{2} \)
good5$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 9 T + 70 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
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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

−13.45671891087735484083981300377, −12.80998468546271284088912910412, −12.07666281357419016843448241277, −11.92664055228089504978083804730, −11.28527064559380212674489377115, −11.23722003571450827227201111174, −10.29710659205747403231092553134, −9.624069631987275098561113869906, −8.811005476416011679890369474546, −8.512409772558547881189167459734, −8.079960609985223141844093130107, −7.76919178146410715595661776394, −7.23421296916661365934298413094, −6.98773585159259695846011777266, −5.23466875149161686678487311624, −4.64558416188840452997018768765, −3.90799930380838217657391987606, −3.90442411212571966814144237123, −2.72349697245102963239984558617, −1.60614571663652286958938868619, 1.60614571663652286958938868619, 2.72349697245102963239984558617, 3.90442411212571966814144237123, 3.90799930380838217657391987606, 4.64558416188840452997018768765, 5.23466875149161686678487311624, 6.98773585159259695846011777266, 7.23421296916661365934298413094, 7.76919178146410715595661776394, 8.079960609985223141844093130107, 8.512409772558547881189167459734, 8.811005476416011679890369474546, 9.624069631987275098561113869906, 10.29710659205747403231092553134, 11.23722003571450827227201111174, 11.28527064559380212674489377115, 11.92664055228089504978083804730, 12.07666281357419016843448241277, 12.80998468546271284088912910412, 13.45671891087735484083981300377

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