Properties

Label 2-159936-1.1-c1-0-119
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 + 3·5-s + 9-s − 11-s + 3·13-s + 3·15-s − 17-s + 6·19-s + 2·23-s + 4·25-s + 27-s − 6·29-s − 33-s − 3·37-s + 3·39-s + 11·43-s + 3·45-s − 51-s + 9·53-s − 3·55-s + 6·57-s + 4·59-s − 14·61-s + 9·65-s − 7·67-s + 2·69-s − 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 0.774·15-s − 0.242·17-s + 1.37·19-s + 0.417·23-s + 4/5·25-s + 0.192·27-s − 1.11·29-s − 0.174·33-s − 0.493·37-s + 0.480·39-s + 1.67·43-s + 0.447·45-s − 0.140·51-s + 1.23·53-s − 0.404·55-s + 0.794·57-s + 0.520·59-s − 1.79·61-s + 1.11·65-s − 0.855·67-s + 0.240·69-s − 1.42·71-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

\(L(1)\) \(\approx\) \(5.411776390\)
\(L(\frac12)\) \(\approx\) \(5.411776390\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−13.35532859874684, −13.12682163721433, −12.34442993370792, −11.99939537019039, −11.21208159069478, −10.86922790547478, −10.32702442029175, −9.908860682967351, −9.288188057101325, −9.120405398189059, −8.638629203132073, −7.943842184071360, −7.355512324956667, −7.115398531212753, −6.267706094998476, −5.852502176216183, −5.501404318227570, −4.902313838260678, −4.216445328507441, −3.602541304049623, −3.000733548225082, −2.541975435825337, −1.821005722764832, −1.406815668149471, −0.6499985546683725, 0.6499985546683725, 1.406815668149471, 1.821005722764832, 2.541975435825337, 3.000733548225082, 3.602541304049623, 4.216445328507441, 4.902313838260678, 5.501404318227570, 5.852502176216183, 6.267706094998476, 7.115398531212753, 7.355512324956667, 7.943842184071360, 8.638629203132073, 9.120405398189059, 9.288188057101325, 9.908860682967351, 10.32702442029175, 10.86922790547478, 11.21208159069478, 11.99939537019039, 12.34442993370792, 13.12682163721433, 13.35532859874684

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