Properties

Label 2-159936-1.1-c1-0-222
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s − 11-s + 2·13-s + 15-s + 17-s − 6·19-s + 6·23-s − 4·25-s + 27-s − 3·29-s − 5·31-s − 33-s + 4·37-s + 2·39-s + 6·41-s + 6·43-s + 45-s − 2·47-s + 51-s − 5·53-s − 55-s − 6·57-s + 3·59-s − 12·61-s + 2·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s + 0.242·17-s − 1.37·19-s + 1.25·23-s − 4/5·25-s + 0.192·27-s − 0.557·29-s − 0.898·31-s − 0.174·33-s + 0.657·37-s + 0.320·39-s + 0.937·41-s + 0.914·43-s + 0.149·45-s − 0.291·47-s + 0.140·51-s − 0.686·53-s − 0.134·55-s − 0.794·57-s + 0.390·59-s − 1.53·61-s + 0.248·65-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: \(1\)
Selberg data: \((2,\ 159936,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.39500054135577, −13.09202301756067, −12.67827183252736, −12.35511492228396, −11.42455535911319, −11.06906994360347, −10.73733643771908, −10.16559096095641, −9.529180086823895, −9.223303210354550, −8.830126432850014, −8.107027038623082, −7.847710290139597, −7.268480491969852, −6.589251181709176, −6.257056831911406, −5.597114234574014, −5.158668571889247, −4.457977318364884, −3.881964641704859, −3.499504599307327, −2.632623329639669, −2.315615736720156, −1.617578562311274, −0.9594003682917409, 0, 0.9594003682917409, 1.617578562311274, 2.315615736720156, 2.632623329639669, 3.499504599307327, 3.881964641704859, 4.457977318364884, 5.158668571889247, 5.597114234574014, 6.257056831911406, 6.589251181709176, 7.268480491969852, 7.847710290139597, 8.107027038623082, 8.830126432850014, 9.223303210354550, 9.529180086823895, 10.16559096095641, 10.73733643771908, 11.06906994360347, 11.42455535911319, 12.35511492228396, 12.67827183252736, 13.09202301756067, 13.39500054135577

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