Properties

Label 2-69-1.1-c5-0-3
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  + 1.42·2-s − 9·3-s − 29.9·4-s − 83.8·5-s − 12.8·6-s + 175.·7-s − 88.5·8-s + 81·9-s − 119.·10-s + 373.·11-s + 269.·12-s + 930.·13-s + 250.·14-s + 755.·15-s + 832.·16-s − 1.10e3·17-s + 115.·18-s − 2.42e3·19-s + 2.51e3·20-s − 1.58e3·21-s + 533.·22-s + 529·23-s + 796.·24-s + 3.91e3·25-s + 1.33e3·26-s − 729·27-s − 5.26e3·28-s + ⋯
L(s)  = 1  + 0.252·2-s − 0.577·3-s − 0.936·4-s − 1.50·5-s − 0.145·6-s + 1.35·7-s − 0.489·8-s + 0.333·9-s − 0.379·10-s + 0.930·11-s + 0.540·12-s + 1.52·13-s + 0.342·14-s + 0.866·15-s + 0.812·16-s − 0.930·17-s + 0.0841·18-s − 1.54·19-s + 1.40·20-s − 0.782·21-s + 0.234·22-s + 0.208·23-s + 0.282·24-s + 1.25·25-s + 0.385·26-s − 0.192·27-s − 1.26·28-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: \(0\)
Selberg data: \((2,\ 69,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.124387034\)
\(L(\frac12)\) \(\approx\) \(1.124387034\)
\(L(\frac{7}{2})\) 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 \( 1 - 1.42T + 32T^{2} \)
5 \( 1 + 83.8T + 3.12e3T^{2} \)
7 \( 1 - 175.T + 1.68e4T^{2} \)
11 \( 1 - 373.T + 1.61e5T^{2} \)
13 \( 1 - 930.T + 3.71e5T^{2} \)
17 \( 1 + 1.10e3T + 1.41e6T^{2} \)
19 \( 1 + 2.42e3T + 2.47e6T^{2} \)
29 \( 1 - 7.81e3T + 2.05e7T^{2} \)
31 \( 1 - 2.72e3T + 2.86e7T^{2} \)
37 \( 1 - 1.28e4T + 6.93e7T^{2} \)
41 \( 1 + 1.22e4T + 1.15e8T^{2} \)
43 \( 1 - 1.26e4T + 1.47e8T^{2} \)
47 \( 1 - 1.48e4T + 2.29e8T^{2} \)
53 \( 1 + 4.38e3T + 4.18e8T^{2} \)
59 \( 1 + 4.00e4T + 7.14e8T^{2} \)
61 \( 1 - 5.52e4T + 8.44e8T^{2} \)
67 \( 1 + 2.98e4T + 1.35e9T^{2} \)
71 \( 1 - 2.80e4T + 1.80e9T^{2} \)
73 \( 1 - 6.34e4T + 2.07e9T^{2} \)
79 \( 1 + 8.00e3T + 3.07e9T^{2} \)
83 \( 1 - 3.95e4T + 3.93e9T^{2} \)
89 \( 1 + 8.44e4T + 5.58e9T^{2} \)
97 \( 1 + 3.24e4T + 8.58e9T^{2} \)
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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

−13.78733439981809371876831295263, −12.48932278333697631523503824702, −11.51090618669035492196798504440, −10.82383021228841810380783455552, −8.733366182979659215949909794024, −8.157660447568575287993811111533, −6.39480949982109111617830612074, −4.57649430467843093323078164884, −4.04508172993792530991655319970, −0.871856112614604151453973578648, 0.871856112614604151453973578648, 4.04508172993792530991655319970, 4.57649430467843093323078164884, 6.39480949982109111617830612074, 8.157660447568575287993811111533, 8.733366182979659215949909794024, 10.82383021228841810380783455552, 11.51090618669035492196798504440, 12.48932278333697631523503824702, 13.78733439981809371876831295263

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