Properties

Label 2-667-1.1-c1-0-46
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.89·2-s − 0.392·3-s + 1.58·4-s − 4.13·5-s − 0.743·6-s + 2.17·7-s − 0.788·8-s − 2.84·9-s − 7.83·10-s − 2.90·11-s − 0.621·12-s + 3.60·13-s + 4.11·14-s + 1.62·15-s − 4.65·16-s − 6.33·17-s − 5.38·18-s − 2.54·19-s − 6.55·20-s − 0.853·21-s − 5.49·22-s − 23-s + 0.309·24-s + 12.1·25-s + 6.82·26-s + 2.29·27-s + 3.44·28-s + ⋯
L(s)  = 1  + 1.33·2-s − 0.226·3-s + 0.791·4-s − 1.84·5-s − 0.303·6-s + 0.821·7-s − 0.278·8-s − 0.948·9-s − 2.47·10-s − 0.875·11-s − 0.179·12-s + 0.999·13-s + 1.09·14-s + 0.419·15-s − 1.16·16-s − 1.53·17-s − 1.26·18-s − 0.582·19-s − 1.46·20-s − 0.186·21-s − 1.17·22-s − 0.208·23-s + 0.0631·24-s + 2.42·25-s + 1.33·26-s + 0.441·27-s + 0.650·28-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: \(1\)
Selberg data: \((2,\ 667,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.89T + 2T^{2} \)
3 \( 1 + 0.392T + 3T^{2} \)
5 \( 1 + 4.13T + 5T^{2} \)
7 \( 1 - 2.17T + 7T^{2} \)
11 \( 1 + 2.90T + 11T^{2} \)
13 \( 1 - 3.60T + 13T^{2} \)
17 \( 1 + 6.33T + 17T^{2} \)
19 \( 1 + 2.54T + 19T^{2} \)
31 \( 1 + 0.623T + 31T^{2} \)
37 \( 1 - 1.79T + 37T^{2} \)
41 \( 1 + 5.90T + 41T^{2} \)
43 \( 1 + 1.37T + 43T^{2} \)
47 \( 1 - 9.54T + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + 6.14T + 59T^{2} \)
61 \( 1 - 0.927T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 - 5.20T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 0.490T + 79T^{2} \)
83 \( 1 + 17.8T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 - 7.45T + 97T^{2} \)
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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

−10.83640345172142218808346902901, −8.664743869117325586470043910270, −8.456553953644733448478392774280, −7.32764094441775499362495463816, −6.28893042611169511617243990056, −5.22746853888916750330038609747, −4.43505416213874845271410972039, −3.74339274849255335300092625221, −2.61920845897709971750264121001, 0, 2.61920845897709971750264121001, 3.74339274849255335300092625221, 4.43505416213874845271410972039, 5.22746853888916750330038609747, 6.28893042611169511617243990056, 7.32764094441775499362495463816, 8.456553953644733448478392774280, 8.664743869117325586470043910270, 10.83640345172142218808346902901

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