Properties

Label 2-160446-1.1-c1-0-25
Degree $2$
Conductor $160446$
Sign $1$
Analytic cond. $1281.16$
Root an. cond. $35.7934$
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  + 2-s + 3-s + 4-s − 4·5-s + 6-s + 2·7-s + 8-s + 9-s − 4·10-s + 12-s + 13-s + 2·14-s − 4·15-s + 16-s + 17-s + 18-s + 4·19-s − 4·20-s + 2·21-s − 6·23-s + 24-s + 11·25-s + 26-s + 27-s + 2·28-s − 4·30-s + 10·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.288·12-s + 0.277·13-s + 0.534·14-s − 1.03·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.894·20-s + 0.436·21-s − 1.25·23-s + 0.204·24-s + 11/5·25-s + 0.196·26-s + 0.192·27-s + 0.377·28-s − 0.730·30-s + 1.79·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160446\)    =    \(2 \cdot 3 \cdot 11^{2} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(1281.16\)
Root analytic conductor: \(35.7934\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{160446} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 160446,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.778858298\)
\(L(\frac12)\) \(\approx\) \(5.778858298\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
11 \( 1 \)
13 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.24190212836186, −12.78549762842972, −12.27485780485367, −11.82683643347978, −11.41959534873894, −11.27664537686374, −10.52578680318593, −9.982158353597886, −9.485051078285522, −8.706477186735284, −8.142326514015190, −8.022819531163040, −7.562358016553593, −7.099634979558196, −6.399228511842473, −5.921236156021143, −5.055539243280185, −4.731368453025468, −4.106772661366034, −3.835590848428691, −3.287536145676056, −2.622015086395693, −2.131530371896412, −1.020527383614924, −0.7363854429394138, 0.7363854429394138, 1.020527383614924, 2.131530371896412, 2.622015086395693, 3.287536145676056, 3.835590848428691, 4.106772661366034, 4.731368453025468, 5.055539243280185, 5.921236156021143, 6.399228511842473, 7.099634979558196, 7.562358016553593, 8.022819531163040, 8.142326514015190, 8.706477186735284, 9.485051078285522, 9.982158353597886, 10.52578680318593, 11.27664537686374, 11.41959534873894, 11.82683643347978, 12.27485780485367, 12.78549762842972, 13.24190212836186

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