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Properties

Label 2-1287-13.12-c1-0-16
Degree $2$
Conductor $1287$
Sign $-0.246 - 0.969i$
Analytic cond. $10.2767$
Root an. cond. $3.20573$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.584i·2-s + 1.65·4-s + 1.95i·5-s − 1.51i·7-s + 2.13i·8-s − 1.14·10-s + i·11-s + (−3.49 + 0.887i)13-s + 0.883·14-s + 2.06·16-s + 3.44·17-s + 5.09i·19-s + 3.23i·20-s − 0.584·22-s − 0.701·23-s + ⋯
L(s)  = 1  + 0.413i·2-s + 0.829·4-s + 0.873i·5-s − 0.571i·7-s + 0.756i·8-s − 0.361·10-s + 0.301i·11-s + (−0.969 + 0.246i)13-s + 0.236·14-s + 0.516·16-s + 0.834·17-s + 1.16i·19-s + 0.724i·20-s − 0.124·22-s − 0.146·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.246 - 0.969i$
Analytic conductor: \(10.2767\)
Root analytic conductor: \(3.20573\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1287,\ (\ :1/2),\ -0.246 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.893475147\)
\(L(\frac12)\) \(\approx\) \(1.893475147\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (3.49 - 0.887i)T \)
good2 \( 1 - 0.584iT - 2T^{2} \)
5 \( 1 - 1.95iT - 5T^{2} \)
7 \( 1 + 1.51iT - 7T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
19 \( 1 - 5.09iT - 19T^{2} \)
23 \( 1 + 0.701T + 23T^{2} \)
29 \( 1 + 5.04T + 29T^{2} \)
31 \( 1 - 7.26iT - 31T^{2} \)
37 \( 1 - 2.08iT - 37T^{2} \)
41 \( 1 + 1.03iT - 41T^{2} \)
43 \( 1 - 5.48T + 43T^{2} \)
47 \( 1 - 3.75iT - 47T^{2} \)
53 \( 1 - 0.502T + 53T^{2} \)
59 \( 1 + 2.65iT - 59T^{2} \)
61 \( 1 + 5.38T + 61T^{2} \)
67 \( 1 + 8.47iT - 67T^{2} \)
71 \( 1 - 2.31iT - 71T^{2} \)
73 \( 1 + 2.21iT - 73T^{2} \)
79 \( 1 + 5.54T + 79T^{2} \)
83 \( 1 + 14.5iT - 83T^{2} \)
89 \( 1 + 1.80iT - 89T^{2} \)
97 \( 1 - 14.5iT - 97T^{2} \)
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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

−10.29651363609618693954306774324, −9.099278501424385379924665974786, −7.79154986544200412565452504468, −7.47184524375912685725831502489, −6.72639641085258225707734131451, −5.94261073601849586519919716758, −4.97631482882452909241522675763, −3.67354534758942577116627068053, −2.77503663042341523356768346008, −1.64485000747622761065290381533, 0.77437542881181959125809743713, 2.14944002976967554942505903918, 2.97002221684973548623295854590, 4.20163710112352242667232381841, 5.31026348483617601895159250136, 5.92039236839240307148506281439, 7.10682536975345966959417261483, 7.73182503885641269649406233374, 8.760982903490531304864030816954, 9.475290637612054270377744557306

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