Properties

Label 2-159936-1.1-c1-0-123
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s − 11-s − 13-s + 15-s − 17-s − 19-s + 5·23-s − 4·25-s − 27-s − 2·29-s − 2·31-s + 33-s + 8·37-s + 39-s − 5·41-s − 5·43-s − 45-s − 12·47-s + 51-s + 55-s + 57-s + 2·59-s + 2·61-s + 65-s − 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s − 0.229·19-s + 1.04·23-s − 4/5·25-s − 0.192·27-s − 0.371·29-s − 0.359·31-s + 0.174·33-s + 1.31·37-s + 0.160·39-s − 0.780·41-s − 0.762·43-s − 0.149·45-s − 1.75·47-s + 0.140·51-s + 0.134·55-s + 0.132·57-s + 0.260·59-s + 0.256·61-s + 0.124·65-s − 0.488·67-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 \( 1 + T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.33723890291711, −13.01811641767198, −12.62329738655532, −11.99249386459425, −11.54777480900443, −11.24527173204127, −10.77310048747070, −10.21208667485851, −9.697098448775391, −9.339426703957879, −8.642452125597169, −8.136669245506554, −7.700444703724849, −7.186961511756313, −6.608802352136144, −6.275526222762634, −5.511498041733336, −5.110697946026066, −4.623880416821329, −4.030526600278771, −3.455329712796425, −2.874515817636044, −2.115671399588663, −1.529501487538492, −0.6570868568538273, 0, 0.6570868568538273, 1.529501487538492, 2.115671399588663, 2.874515817636044, 3.455329712796425, 4.030526600278771, 4.623880416821329, 5.110697946026066, 5.511498041733336, 6.275526222762634, 6.608802352136144, 7.186961511756313, 7.700444703724849, 8.136669245506554, 8.642452125597169, 9.339426703957879, 9.697098448775391, 10.21208667485851, 10.77310048747070, 11.24527173204127, 11.54777480900443, 11.99249386459425, 12.62329738655532, 13.01811641767198, 13.33723890291711

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