Properties

Label 2-671-1.1-c1-0-21
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  + 2.68·2-s − 2.98·3-s + 5.19·4-s − 3.56·5-s − 8.00·6-s + 3.18·7-s + 8.55·8-s + 5.91·9-s − 9.55·10-s − 11-s − 15.4·12-s + 5.51·13-s + 8.54·14-s + 10.6·15-s + 12.5·16-s + 1.59·17-s + 15.8·18-s + 0.342·19-s − 18.4·20-s − 9.51·21-s − 2.68·22-s + 8.05·23-s − 25.5·24-s + 7.69·25-s + 14.7·26-s − 8.70·27-s + 16.5·28-s + ⋯
L(s)  = 1  + 1.89·2-s − 1.72·3-s + 2.59·4-s − 1.59·5-s − 3.26·6-s + 1.20·7-s + 3.02·8-s + 1.97·9-s − 3.02·10-s − 0.301·11-s − 4.47·12-s + 1.52·13-s + 2.28·14-s + 2.74·15-s + 3.14·16-s + 0.385·17-s + 3.73·18-s + 0.0786·19-s − 4.13·20-s − 2.07·21-s − 0.571·22-s + 1.67·23-s − 5.21·24-s + 1.53·25-s + 2.89·26-s − 1.67·27-s + 3.12·28-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

\(L(1)\) \(\approx\) \(2.648594960\)
\(L(\frac12)\) \(\approx\) \(2.648594960\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.68T + 2T^{2} \)
3 \( 1 + 2.98T + 3T^{2} \)
5 \( 1 + 3.56T + 5T^{2} \)
7 \( 1 - 3.18T + 7T^{2} \)
13 \( 1 - 5.51T + 13T^{2} \)
17 \( 1 - 1.59T + 17T^{2} \)
19 \( 1 - 0.342T + 19T^{2} \)
23 \( 1 - 8.05T + 23T^{2} \)
29 \( 1 + 7.64T + 29T^{2} \)
31 \( 1 + 0.725T + 31T^{2} \)
37 \( 1 - 0.0605T + 37T^{2} \)
41 \( 1 + 0.550T + 41T^{2} \)
43 \( 1 + 7.35T + 43T^{2} \)
47 \( 1 + 8.94T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 - 7.06T + 59T^{2} \)
67 \( 1 - 1.98T + 67T^{2} \)
71 \( 1 + 2.76T + 71T^{2} \)
73 \( 1 + 0.581T + 73T^{2} \)
79 \( 1 + 4.50T + 79T^{2} \)
83 \( 1 + 4.28T + 83T^{2} \)
89 \( 1 - 2.73T + 89T^{2} \)
97 \( 1 - 8.34T + 97T^{2} \)
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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

−11.19685225612404933958094864968, −10.58767871514895123393496605606, −8.301033297100918563056084866411, −7.35671815145293535845989533879, −6.70892008466413542877175602168, −5.56932158163445238179313049098, −5.05846794040177683672317749926, −4.23796830439172562208685245238, −3.46864203325557860967842362702, −1.31736535991115388956301080897, 1.31736535991115388956301080897, 3.46864203325557860967842362702, 4.23796830439172562208685245238, 5.05846794040177683672317749926, 5.56932158163445238179313049098, 6.70892008466413542877175602168, 7.35671815145293535845989533879, 8.301033297100918563056084866411, 10.58767871514895123393496605606, 11.19685225612404933958094864968

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