Properties

Label 2-69-1.1-c5-0-6
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.24·2-s + 9·3-s − 26.9·4-s + 53.3·5-s + 20.2·6-s + 89.8·7-s − 132.·8-s + 81·9-s + 120.·10-s + 225.·11-s − 242.·12-s + 725.·13-s + 202.·14-s + 480.·15-s + 564.·16-s + 44.9·17-s + 182.·18-s + 1.21e3·19-s − 1.43e3·20-s + 808.·21-s + 505.·22-s − 529·23-s − 1.19e3·24-s − 274.·25-s + 1.63e3·26-s + 729·27-s − 2.42e3·28-s + ⋯
L(s)  = 1  + 0.397·2-s + 0.577·3-s − 0.842·4-s + 0.955·5-s + 0.229·6-s + 0.693·7-s − 0.732·8-s + 0.333·9-s + 0.379·10-s + 0.560·11-s − 0.486·12-s + 1.19·13-s + 0.275·14-s + 0.551·15-s + 0.551·16-s + 0.0377·17-s + 0.132·18-s + 0.770·19-s − 0.804·20-s + 0.400·21-s + 0.222·22-s − 0.208·23-s − 0.422·24-s − 0.0878·25-s + 0.473·26-s + 0.192·27-s − 0.583·28-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: \(0\)
Selberg data: \((2,\ 69,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.724661241\)
\(L(\frac12)\) \(\approx\) \(2.724661241\)
\(L(\frac{7}{2})\) 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 \( 1 - 2.24T + 32T^{2} \)
5 \( 1 - 53.3T + 3.12e3T^{2} \)
7 \( 1 - 89.8T + 1.68e4T^{2} \)
11 \( 1 - 225.T + 1.61e5T^{2} \)
13 \( 1 - 725.T + 3.71e5T^{2} \)
17 \( 1 - 44.9T + 1.41e6T^{2} \)
19 \( 1 - 1.21e3T + 2.47e6T^{2} \)
29 \( 1 - 2.59e3T + 2.05e7T^{2} \)
31 \( 1 - 585.T + 2.86e7T^{2} \)
37 \( 1 + 3.84e3T + 6.93e7T^{2} \)
41 \( 1 + 4.29e3T + 1.15e8T^{2} \)
43 \( 1 + 2.05e4T + 1.47e8T^{2} \)
47 \( 1 + 5.22e3T + 2.29e8T^{2} \)
53 \( 1 + 1.57e4T + 4.18e8T^{2} \)
59 \( 1 + 1.37e4T + 7.14e8T^{2} \)
61 \( 1 + 1.89e4T + 8.44e8T^{2} \)
67 \( 1 - 1.95e3T + 1.35e9T^{2} \)
71 \( 1 - 7.52e4T + 1.80e9T^{2} \)
73 \( 1 + 4.17e3T + 2.07e9T^{2} \)
79 \( 1 + 9.58e4T + 3.07e9T^{2} \)
83 \( 1 - 1.33e4T + 3.93e9T^{2} \)
89 \( 1 - 5.09e4T + 5.58e9T^{2} \)
97 \( 1 - 4.42e4T + 8.58e9T^{2} \)
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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

−13.87061852647113745508106204674, −13.07196617762348817634379337553, −11.68112475009804447924758796356, −10.10749980356376674184553698368, −9.109096445225030237961047550727, −8.165266536736616001604929017023, −6.25115543635079235497105364240, −4.94715596843835069023723653343, −3.48397817706385910086685121560, −1.47569405733674198145364353982, 1.47569405733674198145364353982, 3.48397817706385910086685121560, 4.94715596843835069023723653343, 6.25115543635079235497105364240, 8.165266536736616001604929017023, 9.109096445225030237961047550727, 10.10749980356376674184553698368, 11.68112475009804447924758796356, 13.07196617762348817634379337553, 13.87061852647113745508106204674

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