Properties

Label 2-1287-13.12-c1-0-31
Degree $2$
Conductor $1287$
Sign $0.204 + 0.978i$
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  − 1.42i·2-s − 0.0397·4-s − 0.0606i·5-s + 1.70i·7-s − 2.79i·8-s − 0.0866·10-s + i·11-s + (3.52 − 0.738i)13-s + 2.43·14-s − 4.07·16-s + 3.75·17-s + 2.02i·19-s + 0.00241i·20-s + 1.42·22-s + 0.704·23-s + ⋯
L(s)  = 1  − 1.00i·2-s − 0.0198·4-s − 0.0271i·5-s + 0.644i·7-s − 0.989i·8-s − 0.0273·10-s + 0.301i·11-s + (0.978 − 0.204i)13-s + 0.651·14-s − 1.01·16-s + 0.910·17-s + 0.464i·19-s + 0.000538i·20-s + 0.304·22-s + 0.146·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.204 + 0.978i$
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: \((2,\ 1287,\ (\ :1/2),\ 0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.007614876\)
\(L(\frac12)\) \(\approx\) \(2.007614876\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.52 + 0.738i)T \)
good2 \( 1 + 1.42iT - 2T^{2} \)
5 \( 1 + 0.0606iT - 5T^{2} \)
7 \( 1 - 1.70iT - 7T^{2} \)
17 \( 1 - 3.75T + 17T^{2} \)
19 \( 1 - 2.02iT - 19T^{2} \)
23 \( 1 - 0.704T + 23T^{2} \)
29 \( 1 - 0.0346T + 29T^{2} \)
31 \( 1 - 1.85iT - 31T^{2} \)
37 \( 1 + 8.82iT - 37T^{2} \)
41 \( 1 + 3.32iT - 41T^{2} \)
43 \( 1 + 5.29T + 43T^{2} \)
47 \( 1 - 6.04iT - 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 - 3.34iT - 59T^{2} \)
61 \( 1 - 3.26T + 61T^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 + 1.96iT - 71T^{2} \)
73 \( 1 + 15.3iT - 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 + 4.19iT - 83T^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 - 5.86iT - 97T^{2} \)
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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

−9.559243618568092007126864642245, −8.940987898960669436926444664321, −7.972230530173257235111686264132, −7.00883317381444504542698022805, −6.10280254389741306116293173296, −5.23699252849555963927390913025, −3.95064077140124545403951598532, −3.18582673760313975159287550798, −2.16787345251671201112128405969, −1.05884031433445127830112777446, 1.19451461927875386340240680812, 2.77257122820474371714405894331, 3.87764780920459698396157904110, 5.01902096894301210844495896425, 5.79925028410368710491061359831, 6.72350348034355632116002523594, 7.15981707996558595074164786457, 8.252375306654125501552103220941, 8.597493705350615521825598296725, 9.799280814696057900366585907138

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