Properties

Label 2-69-1.1-c5-0-9
Degree $2$
Conductor $69$
Sign $-1$
Analytic cond. $11.0664$
Root an. cond. $3.32663$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.2·2-s + 9·3-s + 72.1·4-s − 55.5·5-s − 91.8·6-s + 2.50·7-s − 409.·8-s + 81·9-s + 566.·10-s + 228.·11-s + 649.·12-s + 658.·13-s − 25.5·14-s − 499.·15-s + 1.87e3·16-s − 1.44e3·17-s − 826.·18-s − 982.·19-s − 4.00e3·20-s + 22.5·21-s − 2.33e3·22-s + 529·23-s − 3.68e3·24-s − 42.8·25-s − 6.72e3·26-s + 729·27-s + 180.·28-s + ⋯
L(s)  = 1  − 1.80·2-s + 0.577·3-s + 2.25·4-s − 0.993·5-s − 1.04·6-s + 0.0193·7-s − 2.26·8-s + 0.333·9-s + 1.79·10-s + 0.569·11-s + 1.30·12-s + 1.08·13-s − 0.0348·14-s − 0.573·15-s + 1.82·16-s − 1.21·17-s − 0.601·18-s − 0.624·19-s − 2.23·20-s + 0.0111·21-s − 1.02·22-s + 0.208·23-s − 1.30·24-s − 0.0137·25-s − 1.95·26-s + 0.192·27-s + 0.0435·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-1$
Analytic conductor: \(11.0664\)
Root analytic conductor: \(3.32663\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 69,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
23 \( 1 - 529T \)
good2 \( 1 + 10.2T + 32T^{2} \)
5 \( 1 + 55.5T + 3.12e3T^{2} \)
7 \( 1 - 2.50T + 1.68e4T^{2} \)
11 \( 1 - 228.T + 1.61e5T^{2} \)
13 \( 1 - 658.T + 3.71e5T^{2} \)
17 \( 1 + 1.44e3T + 1.41e6T^{2} \)
19 \( 1 + 982.T + 2.47e6T^{2} \)
29 \( 1 + 7.15e3T + 2.05e7T^{2} \)
31 \( 1 + 9.25e3T + 2.86e7T^{2} \)
37 \( 1 - 2.42e3T + 6.93e7T^{2} \)
41 \( 1 + 4.07e3T + 1.15e8T^{2} \)
43 \( 1 + 1.04e4T + 1.47e8T^{2} \)
47 \( 1 - 9.35e3T + 2.29e8T^{2} \)
53 \( 1 + 3.42e4T + 4.18e8T^{2} \)
59 \( 1 + 7.26e3T + 7.14e8T^{2} \)
61 \( 1 - 2.66e4T + 8.44e8T^{2} \)
67 \( 1 - 5.34e4T + 1.35e9T^{2} \)
71 \( 1 - 2.16e4T + 1.80e9T^{2} \)
73 \( 1 + 8.28e4T + 2.07e9T^{2} \)
79 \( 1 + 2.39e4T + 3.07e9T^{2} \)
83 \( 1 - 8.11e4T + 3.93e9T^{2} \)
89 \( 1 - 1.00e5T + 5.58e9T^{2} \)
97 \( 1 - 3.61e4T + 8.58e9T^{2} \)
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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

−13.02824546645033867602483966886, −11.45777910379020830157247897288, −10.89030682676598167931342723265, −9.357905279675207803957630838114, −8.624749963165132819704580509927, −7.68341097708939397240904285760, −6.58143900012162579823380391934, −3.73197627905230858390345858811, −1.78100448633095974814191542570, 0, 1.78100448633095974814191542570, 3.73197627905230858390345858811, 6.58143900012162579823380391934, 7.68341097708939397240904285760, 8.624749963165132819704580509927, 9.357905279675207803957630838114, 10.89030682676598167931342723265, 11.45777910379020830157247897288, 13.02824546645033867602483966886

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