Properties

Label 2-667-1.1-c1-0-34
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  − 1.08·2-s + 1.50·3-s − 0.824·4-s + 0.786·5-s − 1.63·6-s − 4.63·7-s + 3.06·8-s − 0.720·9-s − 0.852·10-s + 4.80·11-s − 1.24·12-s − 3.44·13-s + 5.02·14-s + 1.18·15-s − 1.67·16-s − 1.46·17-s + 0.780·18-s − 1.81·19-s − 0.648·20-s − 6.99·21-s − 5.20·22-s − 23-s + 4.62·24-s − 4.38·25-s + 3.73·26-s − 5.61·27-s + 3.82·28-s + ⋯
L(s)  = 1  − 0.766·2-s + 0.871·3-s − 0.412·4-s + 0.351·5-s − 0.668·6-s − 1.75·7-s + 1.08·8-s − 0.240·9-s − 0.269·10-s + 1.44·11-s − 0.359·12-s − 0.955·13-s + 1.34·14-s + 0.306·15-s − 0.417·16-s − 0.354·17-s + 0.184·18-s − 0.415·19-s − 0.144·20-s − 1.52·21-s − 1.11·22-s − 0.208·23-s + 0.943·24-s − 0.876·25-s + 0.732·26-s − 1.08·27-s + 0.722·28-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 \( 1 + 1.08T + 2T^{2} \)
3 \( 1 - 1.50T + 3T^{2} \)
5 \( 1 - 0.786T + 5T^{2} \)
7 \( 1 + 4.63T + 7T^{2} \)
11 \( 1 - 4.80T + 11T^{2} \)
13 \( 1 + 3.44T + 13T^{2} \)
17 \( 1 + 1.46T + 17T^{2} \)
19 \( 1 + 1.81T + 19T^{2} \)
31 \( 1 + 4.86T + 31T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 - 9.91T + 41T^{2} \)
43 \( 1 + 6.59T + 43T^{2} \)
47 \( 1 - 2.49T + 47T^{2} \)
53 \( 1 - 1.06T + 53T^{2} \)
59 \( 1 + 8.50T + 59T^{2} \)
61 \( 1 - 5.33T + 61T^{2} \)
67 \( 1 + 2.04T + 67T^{2} \)
71 \( 1 + 7.02T + 71T^{2} \)
73 \( 1 + 6.41T + 73T^{2} \)
79 \( 1 + 4.87T + 79T^{2} \)
83 \( 1 - 1.67T + 83T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + 8.39T + 97T^{2} \)
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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

−9.702470816961914455633612229767, −9.199111563790374966594812019794, −8.792761921763578891589582389330, −7.57157457881064428306585992882, −6.74164577463656626200995976881, −5.73939982090911289722475677101, −4.16573565123540021770021973050, −3.33193960836014992786800795765, −2.00162742897713651363276151262, 0, 2.00162742897713651363276151262, 3.33193960836014992786800795765, 4.16573565123540021770021973050, 5.73939982090911289722475677101, 6.74164577463656626200995976881, 7.57157457881064428306585992882, 8.792761921763578891589582389330, 9.199111563790374966594812019794, 9.702470816961914455633612229767

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