Properties

Label 2-671-1.1-c1-0-22
Degree $2$
Conductor $671$
Sign $1$
Analytic cond. $5.35796$
Root an. cond. $2.31472$
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  + 2.34·2-s − 3.17·3-s + 3.49·4-s + 3.40·5-s − 7.45·6-s − 2.95·7-s + 3.50·8-s + 7.11·9-s + 7.97·10-s + 11-s − 11.1·12-s + 5.60·13-s − 6.93·14-s − 10.8·15-s + 1.23·16-s − 3.44·17-s + 16.6·18-s + 6.93·19-s + 11.8·20-s + 9.40·21-s + 2.34·22-s + 0.836·23-s − 11.1·24-s + 6.57·25-s + 13.1·26-s − 13.0·27-s − 10.3·28-s + ⋯
L(s)  = 1  + 1.65·2-s − 1.83·3-s + 1.74·4-s + 1.52·5-s − 3.04·6-s − 1.11·7-s + 1.24·8-s + 2.37·9-s + 2.52·10-s + 0.301·11-s − 3.20·12-s + 1.55·13-s − 1.85·14-s − 2.79·15-s + 0.307·16-s − 0.834·17-s + 3.92·18-s + 1.59·19-s + 2.65·20-s + 2.05·21-s + 0.499·22-s + 0.174·23-s − 2.27·24-s + 1.31·25-s + 2.57·26-s − 2.51·27-s − 1.95·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $1$
Analytic conductor: \(5.35796\)
Root analytic conductor: \(2.31472\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 671,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.684550956\)
\(L(\frac12)\) \(\approx\) \(2.684550956\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - T \)
61 \( 1 + T \)
good2 \( 1 - 2.34T + 2T^{2} \)
3 \( 1 + 3.17T + 3T^{2} \)
5 \( 1 - 3.40T + 5T^{2} \)
7 \( 1 + 2.95T + 7T^{2} \)
13 \( 1 - 5.60T + 13T^{2} \)
17 \( 1 + 3.44T + 17T^{2} \)
19 \( 1 - 6.93T + 19T^{2} \)
23 \( 1 - 0.836T + 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 - 0.117T + 31T^{2} \)
37 \( 1 + 0.152T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 - 5.87T + 43T^{2} \)
47 \( 1 - 0.889T + 47T^{2} \)
53 \( 1 - 3.79T + 53T^{2} \)
59 \( 1 + 9.37T + 59T^{2} \)
67 \( 1 - 2.66T + 67T^{2} \)
71 \( 1 - 1.13T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 - 5.23T + 79T^{2} \)
83 \( 1 + 17.7T + 83T^{2} \)
89 \( 1 - 2.71T + 89T^{2} \)
97 \( 1 + 16.6T + 97T^{2} \)
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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

−10.77076780255337589561694832458, −10.05762210204428975977393795459, −9.147388959783374889759907544397, −6.85400311597137656938582325522, −6.49625295665763205385490247222, −5.83916156116772364928441013688, −5.35351968755511075212641207212, −4.30332340184491474901980852980, −3.08344965438095343496405755139, −1.39990619534282497521899687499, 1.39990619534282497521899687499, 3.08344965438095343496405755139, 4.30332340184491474901980852980, 5.35351968755511075212641207212, 5.83916156116772364928441013688, 6.49625295665763205385490247222, 6.85400311597137656938582325522, 9.147388959783374889759907544397, 10.05762210204428975977393795459, 10.77076780255337589561694832458

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