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Properties

Label 2-1458-9.7-c1-0-22
Degree $2$
Conductor $1458$
Sign $0.5 + 0.866i$
Analytic cond. $11.6421$
Root an. cond. $3.41206$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.53 + 2.66i)5-s + (0.127 − 0.220i)7-s − 0.999·8-s + 3.07·10-s + (1.72 − 2.99i)11-s + (−2.81 − 4.88i)13-s + (−0.127 − 0.220i)14-s + (−0.5 + 0.866i)16-s + 5.01·17-s − 0.904·19-s + (1.53 − 2.66i)20-s + (−1.72 − 2.99i)22-s + (−2.82 − 4.89i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.688 + 1.19i)5-s + (0.0480 − 0.0832i)7-s − 0.353·8-s + 0.973·10-s + (0.520 − 0.901i)11-s + (−0.782 − 1.35i)13-s + (−0.0339 − 0.0588i)14-s + (−0.125 + 0.216i)16-s + 1.21·17-s − 0.207·19-s + (0.344 − 0.596i)20-s + (−0.368 − 0.637i)22-s + (−0.589 − 1.02i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1458\)    =    \(2 \cdot 3^{6}\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(11.6421\)
Root analytic conductor: \(3.41206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1458} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1458,\ (\ :1/2),\ 0.5 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.268522033\)
\(L(\frac12)\) \(\approx\) \(2.268522033\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
good5 \( 1 + (-1.53 - 2.66i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.127 + 0.220i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.72 + 2.99i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.81 + 4.88i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.01T + 17T^{2} \)
19 \( 1 + 0.904T + 19T^{2} \)
23 \( 1 + (2.82 + 4.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.883 + 1.53i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.32 - 9.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.83T + 37T^{2} \)
41 \( 1 + (-2.85 - 4.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.28 + 7.41i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.26 + 10.8i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.65T + 53T^{2} \)
59 \( 1 + (1.85 + 3.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.88 + 3.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.79 - 6.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.42T + 71T^{2} \)
73 \( 1 + 6.99T + 73T^{2} \)
79 \( 1 + (6.55 - 11.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.46 + 5.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.80T + 89T^{2} \)
97 \( 1 + (4.31 - 7.47i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.765108458968073389315886171109, −8.644990710497560818884797321098, −7.78115359346841863424083299701, −6.76275478859783144101673745564, −5.97620075871321614912365304132, −5.35774571345324787599632121839, −4.06985730054284845411436447960, −2.99370849820331393747076848592, −2.56168284459775077006599120658, −0.931829821947786026922724150740, 1.31322012294842728329271208419, 2.44394221907385955318552184250, 4.17740060733371813208804250991, 4.54092932061845182690857185012, 5.60314515554228153481570924626, 6.16638970505204570958860022592, 7.31123012067473401153064700364, 7.86902969151416194173426655081, 9.017746678126506841454626902291, 9.487393057378062789012963117077

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