Properties

Label 2-287-1.1-c1-0-0
Degree $2$
Conductor $287$
Sign $1$
Analytic cond. $2.29170$
Root an. cond. $1.51383$
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  − 0.644·2-s − 2.95·3-s − 1.58·4-s − 4.36·5-s + 1.90·6-s − 7-s + 2.31·8-s + 5.73·9-s + 2.81·10-s − 4.65·11-s + 4.68·12-s − 0.769·13-s + 0.644·14-s + 12.9·15-s + 1.67·16-s + 0.371·17-s − 3.69·18-s − 1.98·19-s + 6.91·20-s + 2.95·21-s + 3.00·22-s + 1.04·23-s − 6.83·24-s + 14.0·25-s + 0.495·26-s − 8.09·27-s + 1.58·28-s + ⋯
L(s)  = 1  − 0.455·2-s − 1.70·3-s − 0.792·4-s − 1.95·5-s + 0.778·6-s − 0.377·7-s + 0.817·8-s + 1.91·9-s + 0.890·10-s − 1.40·11-s + 1.35·12-s − 0.213·13-s + 0.172·14-s + 3.33·15-s + 0.419·16-s + 0.0901·17-s − 0.871·18-s − 0.456·19-s + 1.54·20-s + 0.645·21-s + 0.640·22-s + 0.217·23-s − 1.39·24-s + 2.81·25-s + 0.0972·26-s − 1.55·27-s + 0.299·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $1$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1100297274\)
\(L(\frac12)\) \(\approx\) \(0.1100297274\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T \)
41 \( 1 - T \)
good2 \( 1 + 0.644T + 2T^{2} \)
3 \( 1 + 2.95T + 3T^{2} \)
5 \( 1 + 4.36T + 5T^{2} \)
11 \( 1 + 4.65T + 11T^{2} \)
13 \( 1 + 0.769T + 13T^{2} \)
17 \( 1 - 0.371T + 17T^{2} \)
19 \( 1 + 1.98T + 19T^{2} \)
23 \( 1 - 1.04T + 23T^{2} \)
29 \( 1 + 9.76T + 29T^{2} \)
31 \( 1 + 7.10T + 31T^{2} \)
37 \( 1 + 0.873T + 37T^{2} \)
43 \( 1 - 9.74T + 43T^{2} \)
47 \( 1 + 6.56T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 + 0.148T + 59T^{2} \)
61 \( 1 - 2.38T + 61T^{2} \)
67 \( 1 + 4.27T + 67T^{2} \)
71 \( 1 + 6.72T + 71T^{2} \)
73 \( 1 - 2.05T + 73T^{2} \)
79 \( 1 + 4.82T + 79T^{2} \)
83 \( 1 - 1.55T + 83T^{2} \)
89 \( 1 + 2.86T + 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−11.62822491702930550348502025978, −10.91218506372194940862045280451, −10.30555528478023615299590518845, −8.938147201607792854376625881176, −7.70289452692794052775321435875, −7.24216761345643382622013783312, −5.59759352229335320574216629958, −4.72943244324318017616117073121, −3.77309444083335479053955210645, −0.38268555086796134873457892850, 0.38268555086796134873457892850, 3.77309444083335479053955210645, 4.72943244324318017616117073121, 5.59759352229335320574216629958, 7.24216761345643382622013783312, 7.70289452692794052775321435875, 8.938147201607792854376625881176, 10.30555528478023615299590518845, 10.91218506372194940862045280451, 11.62822491702930550348502025978

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