Properties

Label 2-270802-1.1-c1-0-0
Degree $2$
Conductor $270802$
Sign $1$
Analytic cond. $2162.36$
Root an. cond. $46.5012$
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 + 2·3-s + 4-s − 2·6-s − 7-s − 8-s + 9-s − 4·11-s + 2·12-s − 2·13-s + 14-s + 16-s − 4·17-s − 18-s − 2·21-s + 4·22-s + 23-s − 2·24-s − 5·25-s + 2·26-s − 4·27-s − 28-s + 10·31-s − 32-s − 8·33-s + 4·34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.577·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.436·21-s + 0.852·22-s + 0.208·23-s − 0.408·24-s − 25-s + 0.392·26-s − 0.769·27-s − 0.188·28-s + 1.79·31-s − 0.176·32-s − 1.39·33-s + 0.685·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270802\)    =    \(2 \cdot 7 \cdot 23 \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(2162.36\)
Root analytic conductor: \(46.5012\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{270802} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 270802,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8867177030\)
\(L(\frac12)\) \(\approx\) \(0.8867177030\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−12.85317800590187, −12.38021497197402, −11.77191325355399, −11.36978438768532, −10.78603951760905, −10.35678790976725, −9.888131565307611, −9.450606789880042, −9.156328172434752, −8.505582634524853, −8.122324935676663, −7.897208002224485, −7.186532838330202, −7.000193963135846, −6.084758190309316, −5.871075959260738, −5.130654230111046, −4.407403475171254, −4.105624461093076, −3.150840376948030, −2.888286057436551, −2.360862135054330, −2.043226831033973, −1.118496024850811, −0.2667825391040626, 0.2667825391040626, 1.118496024850811, 2.043226831033973, 2.360862135054330, 2.888286057436551, 3.150840376948030, 4.105624461093076, 4.407403475171254, 5.130654230111046, 5.871075959260738, 6.084758190309316, 7.000193963135846, 7.186532838330202, 7.897208002224485, 8.122324935676663, 8.505582634524853, 9.156328172434752, 9.450606789880042, 9.888131565307611, 10.35678790976725, 10.78603951760905, 11.36978438768532, 11.77191325355399, 12.38021497197402, 12.85317800590187

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