Properties

Label 2-667-1.1-c1-0-9
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  − 2.22·2-s + 0.328·3-s + 2.93·4-s + 2.84·5-s − 0.729·6-s − 1.80·7-s − 2.08·8-s − 2.89·9-s − 6.32·10-s + 1.40·11-s + 0.964·12-s + 5.09·13-s + 4.00·14-s + 0.934·15-s − 1.24·16-s − 0.501·17-s + 6.42·18-s + 7.81·19-s + 8.35·20-s − 0.591·21-s − 3.13·22-s − 23-s − 0.684·24-s + 3.09·25-s − 11.3·26-s − 1.93·27-s − 5.29·28-s + ⋯
L(s)  = 1  − 1.57·2-s + 0.189·3-s + 1.46·4-s + 1.27·5-s − 0.297·6-s − 0.680·7-s − 0.736·8-s − 0.964·9-s − 1.99·10-s + 0.424·11-s + 0.278·12-s + 1.41·13-s + 1.06·14-s + 0.241·15-s − 0.311·16-s − 0.121·17-s + 1.51·18-s + 1.79·19-s + 1.86·20-s − 0.129·21-s − 0.667·22-s − 0.208·23-s − 0.139·24-s + 0.618·25-s − 2.22·26-s − 0.372·27-s − 0.999·28-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

\(L(1)\) \(\approx\) \(0.8892867945\)
\(L(\frac12)\) \(\approx\) \(0.8892867945\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.22T + 2T^{2} \)
3 \( 1 - 0.328T + 3T^{2} \)
5 \( 1 - 2.84T + 5T^{2} \)
7 \( 1 + 1.80T + 7T^{2} \)
11 \( 1 - 1.40T + 11T^{2} \)
13 \( 1 - 5.09T + 13T^{2} \)
17 \( 1 + 0.501T + 17T^{2} \)
19 \( 1 - 7.81T + 19T^{2} \)
31 \( 1 + 7.79T + 31T^{2} \)
37 \( 1 - 6.33T + 37T^{2} \)
41 \( 1 - 4.39T + 41T^{2} \)
43 \( 1 + 8.99T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 - 8.71T + 53T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 + 7.18T + 61T^{2} \)
67 \( 1 - 9.87T + 67T^{2} \)
71 \( 1 + 0.384T + 71T^{2} \)
73 \( 1 - 9.35T + 73T^{2} \)
79 \( 1 + 4.12T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 - 5.51T + 89T^{2} \)
97 \( 1 - 9.04T + 97T^{2} \)
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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

−10.18534726717053082002440896735, −9.436442228477065328438483353510, −9.064208741518015062528660657665, −8.224047645305079268972768486033, −7.14462514057110105501870016354, −6.18932709977296047843303025425, −5.54658854219835834837691981048, −3.51632012976807128916737389880, −2.30768569437207625735976389202, −1.06157068357453934421540025652, 1.06157068357453934421540025652, 2.30768569437207625735976389202, 3.51632012976807128916737389880, 5.54658854219835834837691981048, 6.18932709977296047843303025425, 7.14462514057110105501870016354, 8.224047645305079268972768486033, 9.064208741518015062528660657665, 9.436442228477065328438483353510, 10.18534726717053082002440896735

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