Properties

Label 2-270802-1.1-c1-0-8
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s − 2·5-s + 2·6-s + 7-s − 8-s + 9-s + 2·10-s − 6·11-s − 2·12-s − 4·13-s − 14-s + 4·15-s + 16-s + 2·17-s − 18-s − 4·19-s − 2·20-s − 2·21-s + 6·22-s + 23-s + 2·24-s − 25-s + 4·26-s + 4·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.80·11-s − 0.577·12-s − 1.10·13-s − 0.267·14-s + 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.436·21-s + 1.27·22-s + 0.208·23-s + 0.408·24-s − 1/5·25-s + 0.784·26-s + 0.769·27-s + 0.188·28-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: \(1\)
Selberg data: \((2,\ 270802,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
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

−12.76171501336546, −12.41631472836800, −11.95679953946480, −11.48129751103156, −11.26731616860684, −10.72819946238592, −10.23437780777451, −10.01998378947686, −9.490757956168060, −8.592329306852397, −8.260636493521090, −7.876096767971233, −7.545720257626819, −6.953771794654649, −6.430606130412660, −5.944004904387205, −5.321407104917694, −4.914090504093759, −4.620900875440099, −3.867798470529452, −3.085553751026631, −2.541134604256920, −2.150336272968780, −1.115620048941077, −0.4818024800287352, 0, 0.4818024800287352, 1.115620048941077, 2.150336272968780, 2.541134604256920, 3.085553751026631, 3.867798470529452, 4.620900875440099, 4.914090504093759, 5.321407104917694, 5.944004904387205, 6.430606130412660, 6.953771794654649, 7.545720257626819, 7.876096767971233, 8.260636493521090, 8.592329306852397, 9.490757956168060, 10.01998378947686, 10.23437780777451, 10.72819946238592, 11.26731616860684, 11.48129751103156, 11.95679953946480, 12.41631472836800, 12.76171501336546

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