Properties

Label 2-71-1.1-c1-0-5
Degree $2$
Conductor $71$
Sign $1$
Analytic cond. $0.566937$
Root an. cond. $0.752952$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.656·2-s + 2.56·3-s − 1.56·4-s − 1.65·5-s + 1.68·6-s − 3.82·7-s − 2.34·8-s + 3.59·9-s − 1.08·10-s + 3.82·11-s − 4.03·12-s + 4·13-s − 2.51·14-s − 4.25·15-s + 1.59·16-s − 3.82·17-s + 2.36·18-s + 5.91·19-s + 2.59·20-s − 9.82·21-s + 2.51·22-s − 3.13·23-s − 6.01·24-s − 2.25·25-s + 2.62·26-s + 1.53·27-s + 6·28-s + ⋯
L(s)  = 1  + 0.464·2-s + 1.48·3-s − 0.784·4-s − 0.740·5-s + 0.688·6-s − 1.44·7-s − 0.828·8-s + 1.19·9-s − 0.343·10-s + 1.15·11-s − 1.16·12-s + 1.10·13-s − 0.671·14-s − 1.09·15-s + 0.399·16-s − 0.927·17-s + 0.557·18-s + 1.35·19-s + 0.581·20-s − 2.14·21-s + 0.535·22-s − 0.654·23-s − 1.22·24-s − 0.451·25-s + 0.515·26-s + 0.296·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(71\)
Sign: $1$
Analytic conductor: \(0.566937\)
Root analytic conductor: \(0.752952\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 71,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.223777872\)
\(L(\frac12)\) \(\approx\) \(1.223777872\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 - T \)
good2 \( 1 - 0.656T + 2T^{2} \)
3 \( 1 - 2.56T + 3T^{2} \)
5 \( 1 + 1.65T + 5T^{2} \)
7 \( 1 + 3.82T + 7T^{2} \)
11 \( 1 - 3.82T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + 3.82T + 17T^{2} \)
19 \( 1 - 5.91T + 19T^{2} \)
23 \( 1 + 3.13T + 23T^{2} \)
29 \( 1 + 3.25T + 29T^{2} \)
31 \( 1 + 3.31T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 - 0.175T + 41T^{2} \)
43 \( 1 + 1.83T + 43T^{2} \)
47 \( 1 + 9.13T + 47T^{2} \)
53 \( 1 + 1.31T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 1.37T + 67T^{2} \)
73 \( 1 - 1.65T + 73T^{2} \)
79 \( 1 - 3.54T + 79T^{2} \)
83 \( 1 - 9.91T + 83T^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 - 4.51T + 97T^{2} \)
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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

−14.50977156469950791808246115204, −13.57065926282305186412372344462, −13.00981412862350610853882012151, −11.66384669158411947055038186581, −9.564728625339990117023664869281, −9.119461665760681099879524635316, −7.913286699958328470371317675007, −6.30708053110179683005478711926, −3.95605268327261641003383631412, −3.36914581499645387434156530848, 3.36914581499645387434156530848, 3.95605268327261641003383631412, 6.30708053110179683005478711926, 7.913286699958328470371317675007, 9.119461665760681099879524635316, 9.564728625339990117023664869281, 11.66384669158411947055038186581, 13.00981412862350610853882012151, 13.57065926282305186412372344462, 14.50977156469950791808246115204

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