Properties

Label 2-160016-1.1-c1-0-0
Degree $2$
Conductor $160016$
Sign $1$
Analytic cond. $1277.73$
Root an. cond. $35.7454$
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  − 4·5-s + 4·7-s − 3·9-s − 4·11-s + 2·17-s + 4·19-s − 4·23-s + 11·25-s − 4·29-s + 2·31-s − 16·35-s + 6·37-s + 6·41-s + 2·43-s + 12·45-s − 6·47-s + 9·49-s − 8·53-s + 16·55-s − 2·61-s − 12·63-s + 4·67-s − 12·71-s + 73-s − 16·77-s − 8·79-s + 9·81-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.51·7-s − 9-s − 1.20·11-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 11/5·25-s − 0.742·29-s + 0.359·31-s − 2.70·35-s + 0.986·37-s + 0.937·41-s + 0.304·43-s + 1.78·45-s − 0.875·47-s + 9/7·49-s − 1.09·53-s + 2.15·55-s − 0.256·61-s − 1.51·63-s + 0.488·67-s − 1.42·71-s + 0.117·73-s − 1.82·77-s − 0.900·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160016\)    =    \(2^{4} \cdot 73 \cdot 137\)
Sign: $1$
Analytic conductor: \(1277.73\)
Root analytic conductor: \(35.7454\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 160016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9413289330\)
\(L(\frac12)\) \(\approx\) \(0.9413289330\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
73 \( 1 - T \)
137 \( 1 - T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
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

−13.19185079293860, −12.73519045308850, −12.11123074308487, −11.75527962221571, −11.42402774866410, −11.04371746197976, −10.70228455425387, −10.03190871066821, −9.344116269921761, −8.762852960836361, −8.135786898157024, −7.978510472002715, −7.692121527394439, −7.286119617507315, −6.426640870553202, −5.629511350645702, −5.380037639871971, −4.719086978796596, −4.348779701409701, −3.755766845174299, −3.023046580672854, −2.742976253197964, −1.851719971139652, −1.057333582600320, −0.3211398770573965, 0.3211398770573965, 1.057333582600320, 1.851719971139652, 2.742976253197964, 3.023046580672854, 3.755766845174299, 4.348779701409701, 4.719086978796596, 5.380037639871971, 5.629511350645702, 6.426640870553202, 7.286119617507315, 7.692121527394439, 7.978510472002715, 8.135786898157024, 8.762852960836361, 9.344116269921761, 10.03190871066821, 10.70228455425387, 11.04371746197976, 11.42402774866410, 11.75527962221571, 12.11123074308487, 12.73519045308850, 13.19185079293860

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