Properties

Label 2-667-1.1-c1-0-36
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  − 1.99·2-s + 0.764·3-s + 1.98·4-s + 2.38·5-s − 1.52·6-s − 0.101·7-s + 0.0328·8-s − 2.41·9-s − 4.76·10-s − 3.79·11-s + 1.51·12-s − 3.75·13-s + 0.201·14-s + 1.82·15-s − 4.03·16-s − 4.24·17-s + 4.81·18-s − 3.15·19-s + 4.73·20-s − 0.0773·21-s + 7.56·22-s − 23-s + 0.0251·24-s + 0.689·25-s + 7.49·26-s − 4.14·27-s − 0.200·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.441·3-s + 0.991·4-s + 1.06·5-s − 0.623·6-s − 0.0382·7-s + 0.0116·8-s − 0.804·9-s − 1.50·10-s − 1.14·11-s + 0.438·12-s − 1.04·13-s + 0.0539·14-s + 0.471·15-s − 1.00·16-s − 1.03·17-s + 1.13·18-s − 0.723·19-s + 1.05·20-s − 0.0168·21-s + 1.61·22-s − 0.208·23-s + 0.00513·24-s + 0.137·25-s + 1.46·26-s − 0.797·27-s − 0.0378·28-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 \( 1 + 1.99T + 2T^{2} \)
3 \( 1 - 0.764T + 3T^{2} \)
5 \( 1 - 2.38T + 5T^{2} \)
7 \( 1 + 0.101T + 7T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 + 3.75T + 13T^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 + 3.15T + 19T^{2} \)
31 \( 1 - 0.905T + 31T^{2} \)
37 \( 1 - 6.40T + 37T^{2} \)
41 \( 1 + 3.03T + 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + 6.38T + 47T^{2} \)
53 \( 1 + 7.07T + 53T^{2} \)
59 \( 1 + 5.17T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 - 14.6T + 71T^{2} \)
73 \( 1 - 6.65T + 73T^{2} \)
79 \( 1 - 7.81T + 79T^{2} \)
83 \( 1 + 9.08T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 + 9.31T + 97T^{2} \)
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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

−9.715464427852257088161101214424, −9.417550508208852806244147186787, −8.391048336560780646785789593826, −7.85718928759285209390848775848, −6.76921118969476574965330549130, −5.74407961815169729440706545706, −4.62507173355967503109310822525, −2.64806495271161451677521196668, −2.05427596932475585611029400552, 0, 2.05427596932475585611029400552, 2.64806495271161451677521196668, 4.62507173355967503109310822525, 5.74407961815169729440706545706, 6.76921118969476574965330549130, 7.85718928759285209390848775848, 8.391048336560780646785789593826, 9.417550508208852806244147186787, 9.715464427852257088161101214424

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