Properties

Label 2-667-1.1-c1-0-7
Degree $2$
Conductor $667$
Sign $1$
Analytic cond. $5.32602$
Root an. cond. $2.30781$
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  − 2.26·2-s − 1.62·3-s + 3.13·4-s + 0.126·5-s + 3.67·6-s + 2.95·7-s − 2.57·8-s − 0.367·9-s − 0.287·10-s + 5.41·11-s − 5.08·12-s − 0.978·13-s − 6.69·14-s − 0.205·15-s − 0.438·16-s + 5.88·17-s + 0.833·18-s − 3.80·19-s + 0.397·20-s − 4.78·21-s − 12.2·22-s − 23-s + 4.17·24-s − 4.98·25-s + 2.21·26-s + 5.46·27-s + 9.25·28-s + ⋯
L(s)  = 1  − 1.60·2-s − 0.936·3-s + 1.56·4-s + 0.0566·5-s + 1.50·6-s + 1.11·7-s − 0.910·8-s − 0.122·9-s − 0.0908·10-s + 1.63·11-s − 1.46·12-s − 0.271·13-s − 1.78·14-s − 0.0531·15-s − 0.109·16-s + 1.42·17-s + 0.196·18-s − 0.873·19-s + 0.0888·20-s − 1.04·21-s − 2.61·22-s − 0.208·23-s + 0.852·24-s − 0.996·25-s + 0.435·26-s + 1.05·27-s + 1.74·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $1$
Analytic conductor: \(5.32602\)
Root analytic conductor: \(2.30781\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 667,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5772571709\)
\(L(\frac12)\) \(\approx\) \(0.5772571709\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + T \)
29 \( 1 - T \)
good2 \( 1 + 2.26T + 2T^{2} \)
3 \( 1 + 1.62T + 3T^{2} \)
5 \( 1 - 0.126T + 5T^{2} \)
7 \( 1 - 2.95T + 7T^{2} \)
11 \( 1 - 5.41T + 11T^{2} \)
13 \( 1 + 0.978T + 13T^{2} \)
17 \( 1 - 5.88T + 17T^{2} \)
19 \( 1 + 3.80T + 19T^{2} \)
31 \( 1 + 3.86T + 31T^{2} \)
37 \( 1 + 8.52T + 37T^{2} \)
41 \( 1 - 1.33T + 41T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 - 3.56T + 47T^{2} \)
53 \( 1 - 9.81T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 - 14.6T + 61T^{2} \)
67 \( 1 - 1.26T + 67T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 8.69T + 83T^{2} \)
89 \( 1 + 3.58T + 89T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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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

−10.54447103924518068323341402704, −9.614441039718662551604719121663, −8.823403006532457491874105602133, −8.056202598946823240766071005428, −7.18896887790897425457288411739, −6.25641001015440631451799313750, −5.31760725951995829147040950855, −4.00446722642356020778785023006, −1.99030016561006839685257089590, −0.903343790852595231006206140851, 0.903343790852595231006206140851, 1.99030016561006839685257089590, 4.00446722642356020778785023006, 5.31760725951995829147040950855, 6.25641001015440631451799313750, 7.18896887790897425457288411739, 8.056202598946823240766071005428, 8.823403006532457491874105602133, 9.614441039718662551604719121663, 10.54447103924518068323341402704

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