Properties

Label 2-69-1.1-c5-0-7
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  − 7.33·2-s − 9·3-s + 21.7·4-s − 0.408·5-s + 65.9·6-s − 4.34·7-s + 74.9·8-s + 81·9-s + 2.99·10-s + 428.·11-s − 195.·12-s + 76.4·13-s + 31.8·14-s + 3.67·15-s − 1.24e3·16-s − 71.8·17-s − 593.·18-s − 1.73e3·19-s − 8.89·20-s + 39.0·21-s − 3.14e3·22-s − 529·23-s − 674.·24-s − 3.12e3·25-s − 560.·26-s − 729·27-s − 94.5·28-s + ⋯
L(s)  = 1  − 1.29·2-s − 0.577·3-s + 0.680·4-s − 0.00730·5-s + 0.748·6-s − 0.0335·7-s + 0.414·8-s + 0.333·9-s + 0.00947·10-s + 1.06·11-s − 0.392·12-s + 0.125·13-s + 0.0434·14-s + 0.00421·15-s − 1.21·16-s − 0.0602·17-s − 0.432·18-s − 1.10·19-s − 0.00497·20-s + 0.0193·21-s − 1.38·22-s − 0.208·23-s − 0.239·24-s − 0.999·25-s − 0.162·26-s − 0.192·27-s − 0.0227·28-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: Trivial
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 \( 1 + 7.33T + 32T^{2} \)
5 \( 1 + 0.408T + 3.12e3T^{2} \)
7 \( 1 + 4.34T + 1.68e4T^{2} \)
11 \( 1 - 428.T + 1.61e5T^{2} \)
13 \( 1 - 76.4T + 3.71e5T^{2} \)
17 \( 1 + 71.8T + 1.41e6T^{2} \)
19 \( 1 + 1.73e3T + 2.47e6T^{2} \)
29 \( 1 + 1.66e3T + 2.05e7T^{2} \)
31 \( 1 + 3.62e3T + 2.86e7T^{2} \)
37 \( 1 + 1.46e3T + 6.93e7T^{2} \)
41 \( 1 - 7.59e3T + 1.15e8T^{2} \)
43 \( 1 - 5.30e3T + 1.47e8T^{2} \)
47 \( 1 + 4.93e3T + 2.29e8T^{2} \)
53 \( 1 - 3.09e3T + 4.18e8T^{2} \)
59 \( 1 + 1.07e4T + 7.14e8T^{2} \)
61 \( 1 + 2.11e4T + 8.44e8T^{2} \)
67 \( 1 + 2.64e4T + 1.35e9T^{2} \)
71 \( 1 + 5.81e4T + 1.80e9T^{2} \)
73 \( 1 - 411.T + 2.07e9T^{2} \)
79 \( 1 + 6.21e4T + 3.07e9T^{2} \)
83 \( 1 + 9.15e4T + 3.93e9T^{2} \)
89 \( 1 + 8.04e4T + 5.58e9T^{2} \)
97 \( 1 - 1.18e5T + 8.58e9T^{2} \)
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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

−13.04406587833282740050242676192, −11.70340133034167438869743182731, −10.76740109164040554298663690158, −9.676436586549930163656831243531, −8.707550233391685252987276532705, −7.40182300731342084116976364585, −6.15563203495115864786021140433, −4.25260793137327661778583783146, −1.61620975891460582248460673660, 0, 1.61620975891460582248460673660, 4.25260793137327661778583783146, 6.15563203495115864786021140433, 7.40182300731342084116976364585, 8.707550233391685252987276532705, 9.676436586549930163656831243531, 10.76740109164040554298663690158, 11.70340133034167438869743182731, 13.04406587833282740050242676192

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