Properties

Label 2-69-1.1-c5-0-14
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  − 1.29·2-s + 9·3-s − 30.3·4-s + 59.1·5-s − 11.6·6-s − 213.·7-s + 80.4·8-s + 81·9-s − 76.2·10-s + 126.·11-s − 273.·12-s − 884.·13-s + 275.·14-s + 532.·15-s + 866.·16-s − 1.17e3·17-s − 104.·18-s − 1.86e3·19-s − 1.79e3·20-s − 1.91e3·21-s − 163.·22-s + 529·23-s + 723.·24-s + 369.·25-s + 1.14e3·26-s + 729·27-s + 6.47e3·28-s + ⋯
L(s)  = 1  − 0.228·2-s + 0.577·3-s − 0.947·4-s + 1.05·5-s − 0.131·6-s − 1.64·7-s + 0.444·8-s + 0.333·9-s − 0.241·10-s + 0.315·11-s − 0.547·12-s − 1.45·13-s + 0.375·14-s + 0.610·15-s + 0.846·16-s − 0.990·17-s − 0.0760·18-s − 1.18·19-s − 1.00·20-s − 0.950·21-s − 0.0719·22-s + 0.208·23-s + 0.256·24-s + 0.118·25-s + 0.331·26-s + 0.192·27-s + 1.55·28-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 \( 1 + 1.29T + 32T^{2} \)
5 \( 1 - 59.1T + 3.12e3T^{2} \)
7 \( 1 + 213.T + 1.68e4T^{2} \)
11 \( 1 - 126.T + 1.61e5T^{2} \)
13 \( 1 + 884.T + 3.71e5T^{2} \)
17 \( 1 + 1.17e3T + 1.41e6T^{2} \)
19 \( 1 + 1.86e3T + 2.47e6T^{2} \)
29 \( 1 + 6.78e3T + 2.05e7T^{2} \)
31 \( 1 + 5.14e3T + 2.86e7T^{2} \)
37 \( 1 - 5.13e3T + 6.93e7T^{2} \)
41 \( 1 - 1.24e4T + 1.15e8T^{2} \)
43 \( 1 - 4.19e3T + 1.47e8T^{2} \)
47 \( 1 - 2.30e4T + 2.29e8T^{2} \)
53 \( 1 - 2.51e4T + 4.18e8T^{2} \)
59 \( 1 + 3.71e4T + 7.14e8T^{2} \)
61 \( 1 - 2.64e4T + 8.44e8T^{2} \)
67 \( 1 + 5.43e4T + 1.35e9T^{2} \)
71 \( 1 - 3.56e4T + 1.80e9T^{2} \)
73 \( 1 - 3.39e4T + 2.07e9T^{2} \)
79 \( 1 + 7.66e4T + 3.07e9T^{2} \)
83 \( 1 + 9.66e4T + 3.93e9T^{2} \)
89 \( 1 - 3.00e4T + 5.58e9T^{2} \)
97 \( 1 + 1.46e4T + 8.58e9T^{2} \)
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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

−13.09940254895752240507338910522, −12.72646068368018810513857599117, −10.40776779568168841346057068582, −9.429406645727139011833165983230, −9.100166308007696524322754992864, −7.22676723287847310542670137545, −5.84392042839921085134309680316, −4.10516298989039327972984346540, −2.37827262495584277283687295674, 0, 2.37827262495584277283687295674, 4.10516298989039327972984346540, 5.84392042839921085134309680316, 7.22676723287847310542670137545, 9.100166308007696524322754992864, 9.429406645727139011833165983230, 10.40776779568168841346057068582, 12.72646068368018810513857599117, 13.09940254895752240507338910522

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