Properties

Label 2-671-1.1-c1-0-24
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  − 2.60·2-s + 2.97·3-s + 4.77·4-s + 0.851·5-s − 7.75·6-s + 3.04·7-s − 7.23·8-s + 5.87·9-s − 2.21·10-s − 11-s + 14.2·12-s + 1.25·13-s − 7.92·14-s + 2.53·15-s + 9.28·16-s − 7.73·17-s − 15.2·18-s + 6.82·19-s + 4.06·20-s + 9.06·21-s + 2.60·22-s + 8.40·23-s − 21.5·24-s − 4.27·25-s − 3.27·26-s + 8.56·27-s + 14.5·28-s + ⋯
L(s)  = 1  − 1.84·2-s + 1.71·3-s + 2.38·4-s + 0.380·5-s − 3.16·6-s + 1.15·7-s − 2.55·8-s + 1.95·9-s − 0.701·10-s − 0.301·11-s + 4.11·12-s + 0.348·13-s − 2.11·14-s + 0.654·15-s + 2.32·16-s − 1.87·17-s − 3.60·18-s + 1.56·19-s + 0.909·20-s + 1.97·21-s + 0.555·22-s + 1.75·23-s − 4.40·24-s − 0.855·25-s − 0.641·26-s + 1.64·27-s + 2.74·28-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

\(L(1)\) \(\approx\) \(1.434699284\)
\(L(\frac12)\) \(\approx\) \(1.434699284\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.60T + 2T^{2} \)
3 \( 1 - 2.97T + 3T^{2} \)
5 \( 1 - 0.851T + 5T^{2} \)
7 \( 1 - 3.04T + 7T^{2} \)
13 \( 1 - 1.25T + 13T^{2} \)
17 \( 1 + 7.73T + 17T^{2} \)
19 \( 1 - 6.82T + 19T^{2} \)
23 \( 1 - 8.40T + 23T^{2} \)
29 \( 1 - 1.09T + 29T^{2} \)
31 \( 1 + 2.03T + 31T^{2} \)
37 \( 1 + 6.32T + 37T^{2} \)
41 \( 1 + 9.82T + 41T^{2} \)
43 \( 1 + 8.23T + 43T^{2} \)
47 \( 1 - 1.11T + 47T^{2} \)
53 \( 1 + 4.72T + 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
67 \( 1 - 6.91T + 67T^{2} \)
71 \( 1 + 7.00T + 71T^{2} \)
73 \( 1 - 1.27T + 73T^{2} \)
79 \( 1 + 3.71T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 + 3.62T + 89T^{2} \)
97 \( 1 + 0.113T + 97T^{2} \)
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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

−10.11247617120883274978548505092, −9.368983541366502148078643964522, −8.682121105489843642237222553735, −8.317892571957211476011345106899, −7.39520946576578936140901741198, −6.80705890993076178194043429334, −5.01629584045294775542968488282, −3.31834160293547262233947083900, −2.22326558543756759133381733443, −1.48630621870212293968634266202, 1.48630621870212293968634266202, 2.22326558543756759133381733443, 3.31834160293547262233947083900, 5.01629584045294775542968488282, 6.80705890993076178194043429334, 7.39520946576578936140901741198, 8.317892571957211476011345106899, 8.682121105489843642237222553735, 9.368983541366502148078643964522, 10.11247617120883274978548505092

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