Properties

Label 2-159936-1.1-c1-0-1
Degree $2$
Conductor $159936$
Sign $1$
Analytic cond. $1277.09$
Root an. cond. $35.7364$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s − 6·11-s + 15-s − 17-s + 3·19-s − 7·23-s − 4·25-s + 27-s − 6·29-s − 8·31-s − 6·33-s − 7·37-s − 9·43-s + 45-s − 6·47-s − 51-s − 2·53-s − 6·55-s + 3·57-s + 11·59-s − 6·61-s + 9·67-s − 7·69-s + 9·71-s + 12·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.80·11-s + 0.258·15-s − 0.242·17-s + 0.688·19-s − 1.45·23-s − 4/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 1.04·33-s − 1.15·37-s − 1.37·43-s + 0.149·45-s − 0.875·47-s − 0.140·51-s − 0.274·53-s − 0.809·55-s + 0.397·57-s + 1.43·59-s − 0.768·61-s + 1.09·67-s − 0.842·69-s + 1.06·71-s + 1.40·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(159936\)    =    \(2^{6} \cdot 3 \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(1277.09\)
Root analytic conductor: \(35.7364\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{159936} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 159936,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4225504119\)
\(L(\frac12)\) \(\approx\) \(0.4225504119\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
17 \( 1 + T \)
good5 \( 1 - T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.29683389100455, −12.97781201001086, −12.41934129148817, −11.95708121685851, −11.23813165904526, −10.91673363672158, −10.26255808116891, −9.945783090618785, −9.497588626324659, −9.061368899981436, −8.204086224678035, −8.038334510890517, −7.660764476217803, −6.920323034942579, −6.561963704036485, −5.626056371343228, −5.389462119572029, −5.055528942064243, −4.096481717093971, −3.639576421897300, −3.151201121173965, −2.281397597129547, −2.096376019194111, −1.435772960899365, −0.1661620008147347, 0.1661620008147347, 1.435772960899365, 2.096376019194111, 2.281397597129547, 3.151201121173965, 3.639576421897300, 4.096481717093971, 5.055528942064243, 5.389462119572029, 5.626056371343228, 6.561963704036485, 6.920323034942579, 7.660764476217803, 8.038334510890517, 8.204086224678035, 9.061368899981436, 9.497588626324659, 9.945783090618785, 10.26255808116891, 10.91673363672158, 11.23813165904526, 11.95708121685851, 12.41934129148817, 12.97781201001086, 13.29683389100455

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