Properties

Label 2-1287-1.1-c1-0-35
Degree $2$
Conductor $1287$
Sign $-1$
Analytic cond. $10.2767$
Root an. cond. $3.20573$
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  − 2.14·2-s + 2.59·4-s − 0.447·5-s + 4.87·7-s − 1.26·8-s + 0.958·10-s − 11-s − 13-s − 10.4·14-s − 2.47·16-s − 6.61·17-s − 5.98·19-s − 1.15·20-s + 2.14·22-s + 0.694·23-s − 4.79·25-s + 2.14·26-s + 12.6·28-s − 2.14·29-s − 2.44·31-s + 7.82·32-s + 14.1·34-s − 2.18·35-s + 0.921·37-s + 12.8·38-s + 0.565·40-s − 1.12·41-s + ⋯
L(s)  = 1  − 1.51·2-s + 1.29·4-s − 0.200·5-s + 1.84·7-s − 0.446·8-s + 0.303·10-s − 0.301·11-s − 0.277·13-s − 2.79·14-s − 0.617·16-s − 1.60·17-s − 1.37·19-s − 0.259·20-s + 0.456·22-s + 0.144·23-s − 0.959·25-s + 0.420·26-s + 2.38·28-s − 0.398·29-s − 0.439·31-s + 1.38·32-s + 2.43·34-s − 0.369·35-s + 0.151·37-s + 2.07·38-s + 0.0894·40-s − 0.175·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-1$
Analytic conductor: \(10.2767\)
Root analytic conductor: \(3.20573\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1287,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
11 \( 1 + T \)
13 \( 1 + T \)
good2 \( 1 + 2.14T + 2T^{2} \)
5 \( 1 + 0.447T + 5T^{2} \)
7 \( 1 - 4.87T + 7T^{2} \)
17 \( 1 + 6.61T + 17T^{2} \)
19 \( 1 + 5.98T + 19T^{2} \)
23 \( 1 - 0.694T + 23T^{2} \)
29 \( 1 + 2.14T + 29T^{2} \)
31 \( 1 + 2.44T + 31T^{2} \)
37 \( 1 - 0.921T + 37T^{2} \)
41 \( 1 + 1.12T + 41T^{2} \)
43 \( 1 - 1.14T + 43T^{2} \)
47 \( 1 + 8.67T + 47T^{2} \)
53 \( 1 - 5.01T + 53T^{2} \)
59 \( 1 + 7.47T + 59T^{2} \)
61 \( 1 + 3.85T + 61T^{2} \)
67 \( 1 + 4.73T + 67T^{2} \)
71 \( 1 + 8.60T + 71T^{2} \)
73 \( 1 - 1.77T + 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 - 4.57T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−8.937414879889722070460737140112, −8.586424222629073007647324989130, −7.81195856424267755427453336871, −7.27229295978303115801710938582, −6.19270032234716683916446233003, −4.86448573532046726081695247724, −4.25221996259724188293574580785, −2.28452873738212205448184262545, −1.66404261003355882042009979987, 0, 1.66404261003355882042009979987, 2.28452873738212205448184262545, 4.25221996259724188293574580785, 4.86448573532046726081695247724, 6.19270032234716683916446233003, 7.27229295978303115801710938582, 7.81195856424267755427453336871, 8.586424222629073007647324989130, 8.937414879889722070460737140112

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